The Handbook of Essential Mathematics

Transcription

1 Fo Puic Relese: Distiutio Ulimited The Ai Foce Resech Lotoy The Hdook of Essetil Mthemtics Fomuls, Pocesses, d Tles Plus Applictios i Pesol Fice X Y Y XY Y X X XY X Y X XY Y Compiltio d Epltios: Joh C. Spks Editos: Dold D. Gegoy d Vicet R. Mille Fo Puic Relese: Distiutio Ulimited

2 The Hdook of Essetil Mthemtics Ai Foce Pulictio 6 Fo Pulic Relese: Distiutio Ulimited

3 Fowd Wight-Ptteso Ai Foce Bse WPAFB hs ejoyed legthy d distiguished histoy of sevig the Gete-Dyto commuity i viety of wys. Oe of these wys is though the WPAFB Eductiol Outech EO Pogm, fo which the Ai Foce Resech Lotoy AFRL is poud d cotiuous suppote, povidig oth techicl epetise fom ove pcticig scietists d egiees d ogoig esouces fo the vious pogms sposoed y the WPAFB Eductiol Outech. The missio of the WPAFB EO pogm is To fom leig pteships with the K- eductiol commuity i ode to icese studet weess d ecitemet i ll fields of mth, sciece, vitio, d eospce; ultimtely developig ou tio s futue scietific d techicl wokfoce. I suppot of this missio, the WPAFB EO spies to e the est oe-stop esouce fo ecougemet d ehcemet of K- sciece, mth d techology eductio thoughout the Uited Sttes Ai Foce. It is i this spiit tht AFRL offes The Hdook of Essetil Mthemtics, compedium of mthemticl fomuls d othe useful techicl ifomtio tht will well seve oth studets d teches like fom ely gdes though ely college. It is ou sicee hope tht you will use this esouce to eithe futhe you ow eductio o the eductio of those futue scietists d egiees so vitl to pesevig ou cheished Ameic feedoms. LESTER MCFAWN, SES Eecutive Diecto Ai Foce Resech Lotoy 3

4 Itoductio Fomuls! They seem to e the e of evey egiig mthemtics studet who hs yet to elize tht fomuls e out stuctue d eltioship d ot out memoiztio. Gted, fomuls hve to e memoized; fo, it is ptly though memoiztio tht we evetully ecome ucosciously competet : tue mste of ou skill, pcticig it i lmost effotless, utomtic sese. I mthemtics, eig ucosciously competet mes we hve msteed the udelyig lgeic lguge to the sme degee tht we hve msteed ou tive togue. Kowig fomuls d udestdig the esoig ehid them popels oe towds the od to mthemticl fluecy, so essetil i ou mode high-tech society. The Hdook of Essetil Mthemtics cotis thee mjo sectios. Sectio I, Fomuls, cotis most of the mthemticl fomuls tht peso would epect to ecoute though the secod ye of college egdless of mjo. I dditio, thee e fomuls ely see i such compiltios, icluded s mthemticl tet fo the iquisitive. Sectio I lso icludes select mthemticl pocesses, such s the pocess fo solvig lie equtio i oe ukow, with suppotig emples. Sectio II, Tles, icludes oth pue mth tles d physicl-sciece tles, useful i viety of disciples gig fom physics to usig. As i Sectio I, some tles e icluded just to utue cuiosity i spiit of fu. I Sectios I d II, ech fomul d tle is eumeted fo esy efel. Sectio III, Applictios i Pesol Fice, is smll tetook withi ook whee the lguge of lge is pplied to tht eveydy ficil wold ffectig ll of us thoughout ou lives fom ith to deth. Note: The ide of comiig mthemtics fomuls with ficil pplictios is ot oigil i tht my fthe hd simil type ook s Pudue egieeig studet i the ely 93s. I would like to tke this oppotuity to thk M. Al Gimoe Chim of the Deptmet of Mthemtics, Sicli Commuity College, Dyto, Ohio fo povidig equiedmemoiztio fomul lists fo Sicli mthemtics couses fom which the fomul compiltio ws ptilly uilt. Joh C. Spks Mch 6 4

5 Dedictio The Hdook of Essetil Mthemtics is dedicted to ll Ai Foce fmilies O Icus I ide high... With whoosh to my ck Ad o wid to my fce, Folded hds I quiet est Wtchig...O Icus... The clouds glide y, Thei fields f elow Of gold-illumed sow, Ple yellow, tquil moo To my ight Eveig sky. Ad Wight...O Icus... Mde it so Silveed chiot stekig O togues of fie lepig Ad I will soo e sleepig Aove you dems... August : Joh C. Spks th Aivesy of Poweed Flight

6 Tle of Cotets Sectio I: Fomuls with Select Pocesses Ide to Pocesses Pge 6. Alge 3.. Wht is Vile? 3.. Field Aioms 4.3. Divisiility Tests 5.4. Sutctio, Divisio, Siged Numes 6.5. Rules fo Fctios 8.6. Ptil Fctios 9.7. Rules fo Epoets.8. Rules fo Rdicls.9. Fcto Fomuls.. Lws of Equlity 4.. Lws of Iequlity 6.. Ode of Opetios 7.3. Thee Meigs of Equls 7.4. The Seve Petheses Rules 8.5. Rules fo Logithms 3.6. Comple Numes 3.7. Wht is Fuctio? 3.8. Fuctio Alge Qudtic Equtios & Fuctios 34.. Cdo s Cuic Solutio 36.. Theoy of Polyomil Equtios 37.. Detemits d Cme s Rule Biomil Theoem 4.4. Aithmetic Seies 4.5. Geometic Seies 4.6. Boole Alge 4.7. Vitio Fomuls 43 6

7 Tle of Cotets cot. Clssicl d Alytic Geomety 44.. The Pllel Postultes 44.. Agles d Lies Tigles Coguet Tigles Simil Tigles Pl Figues Solid Figues Pythgoe Theoem 5.9. Heo s Fomul 5.. Golde Rtio 53.. Distce d Lie Fomuls 54.. Fomuls fo Coic Sectios Coic Sectios 3. Tigoomety Bsic Defiitios: Fuctios & Iveses Fudmetl Defiitio-Bsed Idetities Pythgoe Idetities Negtive Agle Idetities Sum d Diffeece Idetities Doule Agle Idetities Hlf Agle Idetities Geel Tigle Fomuls Ac d Secto Fomuls Degee/Rdi Reltioship Additio of Sie d Cosie Pol Fom of Comple Numes Rectgul to Pol Coodites Tigoometic Vlues fom Right Tigles Elemety Vecto Alge Bsic Defiitios d Popeties Dot Poducts Coss Poducts Lie d Ple Equtios Miscelleous Vecto Equtios 67 7

8 Tle of Cotets cot 5. Elemety Clculus Wht is Limit? Wht is Diffeetil? Bsic Diffeetitio Rules Tscedetl Diffeetitio Bsic Atidiffeetitio Rules Tscedetl Atidiffeetitio Lies d Appoimtio Itepettio of Defiite Itegl Fudmetl Theoem of Clculus Geometic Itegl Fomuls Select Elemety Diffeetil Equtios Lplce Tsfom: Geel Popeties Lplce Tsfom: Specific Tsfoms Moey d Fice Wht is Iteest? Simple Iteest Compoud d Cotiuous Iteest Effective Iteest Rtes Peset-to-Futue Vlue Fomuls Peset Vlue of Futue Deposit Stem Peset Vlue of with Iitil Lump Sum Peset Vlue of Cotiuous Types o Retiemet Svigs Accouts Lo Amotiztio Auity Fomuls Mkup d Mkdow Clculus of Fice Poility d Sttistics Poility Fomuls Bsic Cocepts of Sttistics Mesues of Cetl Tedecy Mesues of Dispesio Smplig Distiutio of the Me Smplig Distiutio of the Popotio 9 8

9 Tle of Cotets cot Sectio II: Tles. Numeicl 94.. Fctos of Iteges though Pime Numes less th Rom Numel d Aic Equivlets Nie Elemety Memoy Numes Ameic Nmes fo Lge Numes Selected Mgic Sques Thitee-y-Thitee Multiplictio Tle.8. The Rdom Digits of PI.9. Stdd Noml Distiutio 3.. Two-Sided Studet s t Distiutio 4.. Dte d Dy of Ye 5. Physicl Scieces 6.. Covesio Fctos i Allied Helth 6.. Medicl Aevitios i Allied Helth 7.3. Wid Chill Tle 8.4. Het Ide Tle 8.5. Tempetue Covesio Fomuls 9.6. Uit Covesio Tle 9.7. Popeties of Eth d Moo.8. Metic System 3.9. Bitish System 4 Sectio III: Applictios i Pesol Fice. The Alge of Iteest 8.. Wht is Iteest? 8.. Simple Iteest.3. Compoud Iteest.4. Cotiuous Iteest 4.5. Effective Iteest Rte 9 9

10 Tle of Cotets cot. The Alge of the Nest Egg 35.. Peset d Futue Vlue 35.. Gowth of Iitil Lump Sum Deposit Gowth of Deposit Stem 4.4. The Two Gowth Mechisms i Cocet Summy 5 3. The Alge of Cosume Det Lo Amotiztio You Home Motgge C Los d Leses The Auity s Motgge i Revese The Clculus of Fice Jco Beoulli s Diffeetil Equtio Diffeetils d Iteest Rte Beoulli d Moey Applictios 9 Appedices A. Geek Alphet B. Mthemticl Symols C. My Most Used Fomuls 4

11 Sectio I Fomuls with Select Pocesses

12 Ide to Pocesses Pocess Whee i Sectio I. Comple Rtioliztio Pocess.8.. Qudtic Tiomil Fctoig Pocess Lie Equtio Solutio Pocess.. 4. Lie Iequlity Solutio Pocess Ode of Opetios.. 6. Ode of Opetios with Petheses Rules Logithmic Simplifictio Pocess Comple Nume Multiplictio Comple Nume Divisio.6.9. Pocess of Costuctig Ivese Fuctios.8.6. Qudtic Equtios y Fomul.9.3. Qudtic Equtios y Fctoig Cdo s Cuic Solutio Pocess Cme s Rule, Two-y-Two System Cme s Rule, Thee-y-Thee System Removl of y Tem i Coic Sectios The Lie Fist-Ode Diffeetil Equtio Medi Clcultio 7.3.6

13 . Alge.. Wht is Vile? I the fll of 96, I fist ecouteed the moste clled i my high-school feshm lge clss. The lette is still moste to my, whose el tue hs ee cofused y such wods s vile d ukow: pehps the most hoifyig desciptio of eve iveted! Actully, is vey esily udestood i tems of lguge metpho. I Eglish, we hve oth pope ous d poous whee oth e distict d diffeet pts of speech. Pope ous e specific pesos, plces, o thigs such s Joh, Ohio, d Toyot. Poous e ospecific pesos o etities such s he, she, o it. To see how the cocept of poous d ous pplies to lge, we fist emie ithmetic, which c e thought of s pecise lguge of qutifictio hvig fou ctio ves, ve of eig, d pletho of pope ous. The fou ctio ves e dditio, sutctio, multiplictio, d divisio deoted espectively y,,,. The ve of eig is clled equls o is, 3 deoted y. Specific umes such s, 3. 45, 3 5, 3 769,.4563, 45,, seve s the ithmeticl equivlet to pope ous i Eglish. So, wht is? is meely ospecific ume, the mthemticl equivlet to poou i Eglish. Eglish poous getly epd ou cpility to descie d ifom i geel fshio. Hece, poous dd icesed fleiility to the Eglish lguge. Likewise, mthemticl poous such s, y, z, see Appedi B fo list of symols used i this ook getly epd ou cpility to qutify i geel fshio y ddig fleiility to ou lguge of ithmetic. Aithmetic, with the dditio of, y, z d othe mthemticl poous s ew pt of speech, is clled lge. I Summy: Alge c e defied s geelized ithmetic tht is much moe poweful d fleile th stdd ithmetic. The icesed cpility of lge ove ithmetic is due to the iclusio of the mthemticl poou d its ssocites y, z, etc. A moe use-fiedly me fo vile o ukow is poume. 3

14 .. Field Aioms The field ioms decee the fudmetl opetig popeties of the el ume system d povide the sis fo ll dvced opetig popeties i mthemtics. Let, & c e y thee el umes poumes. The field ioms e s follows. Popeties Additio Multiplictio. Closue is uique el ume is uique el ume Commuttive Associtive c c c c Idetity Ivese Distiutive o Likig Popety Tsitivity Note: c c & c c > & > c > c < & < c < c e ltete epesettios of 4

15 .3. Divisiility Tests Diviso Coditio Tht Mkes it So The lst digit is,,4,6, o 8 3 The sum of the digits is divisile y 3 4 The lst two digits e divisile y 4 5 The lst digit is o 5 6 The ume is divisile y oth d 3 The ume fomed y ddig five times the lst 7 digit to the ume defied y the emiig digits is divisile y 7** 8 The lst thee digits e divisile y 8 9 The sum of the digits is divisile y 9 The lst digit is divides the ume fomed y sutctig two times the lst digit fom the emiig digits** The ume is divisile y oth 3 d divides the ume fomed y ddig fou times the lst digit to the emiig digits** 4 The ume is divisile y oth d 7 5 The ume is divisile y oth 3 d divides the ume fomed y sutctig five times the lst digit fom the emiig digits** 9 9 divides the ume fomed y ddig two times the lst digit to the emiig digits** 3 3 divides the ume fomed y ddig seve times the lst digit to the emiig digits** 9 9 divides the ume fomed y ddig thee times the lst digit to the emiig digits** 3 divides the ume fomed y sutctig 3 thee times the lst digit fom the emiig digits** 37 divides the ume fomed y sutctig 37 eleve times the lst digit fom the emiig digits** **These tests e itetive tests i tht you cotiue to cycle though the pocess util ume is fomed tht c e esily divided y the diviso i questio. 5

16 .4. Sutctio, Divisio, Siged Numes.4.. Defiitios: Sutctio: Divisio:.4.. Altete epesettio of :.4.3. Divisio Popeties of Zeo Zeo i umeto: Zeo i deomito: is udefied Zeo i oth: is udefied.4.4. Demosttio tht divisio-y-zeo is udefied c c fo ll el umes c, the c fo ll el umes, If lgeic impossiility.4.5. Demosttio tht ttempted divisio-y-zeo leds to eoeous esults. Let y y y ; the multiplyig oth sides y gives y y y y y y Dividig oth sides y y whee y gives y y y y. The lst equlity is flse sttemet. 6

17 .4.6. Siged Nume Multiplictio:.4.7. Tle fo Multiplictio of Siged Numes: the itlicized wods i the ody of the tle idicte the esultig sig of the ssocited poduct. Multiplictio of Sig of Sig of Plus Mius Plus Plus Mius Mius Mius Plus.4.8. Demosttio of the lgeic esoleess of the lws of multiplictio fo siged umes. I oth colums, oth the middle d ightmost umes decese i the epected logicl fshio

18 .5. Rules fo Fctios c Let d e fctios with d d. d.5.. Fctiol Equlity: d c.5.. Fctiol Equivlecy:.5.3. Additio like deomitos:.5.4. Additio ulike deomitos: c d c c c c c d c d c d d d d Note: d is the commo deomito.5.5. Sutctio like deomitos:.5.6. Sutctio ulike deomitos: c d c d.5.7. Multiplictio:.5.8. Divisio:.5.9. Divisio missig qutity:.5.. Reductio of Comple Fctio:.5.. Plcemet of Sig: c c c c c c d d d d c c d d c d d c d c c c c c c d d c d c c 8

19 9.6. Ptil Fctios Let P e polyomil epessio with degee less th the degee of the fctoed deomito s show..6.. Two Distict Lie Fctos: B A P The umetos B A, e give y P B P A,.6.. Thee Distict Lie Fctos: c C B A c P The umetos C B A,, e give y,, c c c P C c P B c P A.6.3. N Distict Lie Fctos: i i i i i A P with i j j j i i i P A

20 .7. Rules fo Epoets.7.. Additio: m m.7.. Sutctio: m m.7.3. Multiplictio: m m.7.4. Distiuted ove Simple Poduct:.7.5. Distiuted ove Comple Poduct: p m p m.7.6. Distiuted ove Simple Quotiet:.7.7. Distiuted ove Comple Quotiet: p m p m.7.8. Defiitio of Negtive Epoet:.7.9. Defiitio of Rdicl Epessio:.7.. Defiitio whe No Epoet is Peset:.7.. Defiitio of Zeo Epoet:.7.. Demosttio of the lgeic esoleess of the defiitios fo d vi successive divisios y. Notice the powe deceses y with ech divisio [] [] []

21 .8. Rules fo Rdicls.8.. Bsic Defiitios: d.8.. Comple Rdicl: m.8.3. Associtive:.8.4. Simple Poduct: m m m.8.5. Simple Quotiet:.8.6. Comple Poduct: m m.8.7. Comple Quotiet: m m.8.8. Nestig: m.8.9. Rtiolizig Numeto fo > m :.8.. Rtiolizig Deomito fo > m :.8.. Comple Rtioliztio Pocess: c c c c c c c m m m m m m m m Numeto: c c c.8.. Defiitio of Sud Pis: If ± the the ssocited sud is give y is dicl epessio, m.

22 .9. Fcto Fomuls.9.. Simple Commo Fcto: c c c.9.. Gouped Commo Fcto: d c d c c c d c dc d c.9.3. Diffeece of Sques:.9.4. Epded Diffeece of Sques: c c c.9.5. Sum of Sques: i i i comple.9.6. Pefect Sque: ± ±.9.7. Geel Tiomil:.9.8. Sum of Cues: Diffeece of Cues: Diffeece of Fouths: Powe Reductio to Itege: Powe Reductio to Rdicl:.9.3. Powe Reductio to Itege plus Rdicl:

23 .9.4. Qudtic Tiomil Fctoig Pocess c Let e qudtic tiomil whee the thee coefficiets,, c e iteges. M, N such tht Step : Fid iteges M N. M N c Step : Sustitute fo, c M N c Step 3: Fcto y Goupig.9. M N c M N M N M M N M N Note: if thee e o pi of iteges M, N with oth M N d M N c the the qudtic tiomil is pime. Emple: Fcto the epessio 3 7. : MN 7 4 & M N 3 M 4, N : :

24 .. Lws of Equlity Let A B e lgeic equlity dc, D e y qutities.... Additio: A C B C... Sutctio: A C B C..3. Multiplictio: A C B C..4. Divisio: A B povided C C C..5. Epoet: A B povided is itege..6. Recipocl: povided A, B A B C D..7. Mes & Etemes: CB AD if A, B A B..8. Zeo Poduct Popety: A B A o B..9. The Cocept of Equivlecy Whe solvig equtios, the Lws of Equlity with the eceptio of..5, which poduces equtios with et o eteous solutios i dditio to those fo the oigil equtio e used to mufctue equtios tht e equivlet to the oigil equtio. Equivlet equtios e equtios tht hve ideticl solutios. Howeve, equivlet equtios e ot ideticl i ppece. The gol of y equtio-solvig pocess is to use the Lws of Equlity to cete successio of equivlet equtios whee ech equtio i the equivlecy chi is lgeiclly simple th the pecedig oe. The fil equtio i the chi should e epessio of the fom, the o-ie fom tht llows the solutio to e immeditely detemied. I tht lgeic mistkes c e mde whe poducig the equivlecy chi, the fil swe must lwys e checked i the oigil equtio. Whe usig..5, oe must check fo eteous solutios d delete them fom the solutio set.... Lie Equtio Solutio Pocess Stt with the geel fom L R whee L d R e fist-degee polyomil epessios o the left-hd side d ight-hd side of the equls sig. 4

25 Step : Usig pope lge, idepedetly comie like tems fo oth L d R Step : Use.. d.. o s-eeded sis to cete equivlet equtio of the fom. Step 3: use eithe..3 o..4 to cete the fil equivlet fom fom which the solutio is esily deduced. Step 4: Check solutio i oigil equtio. Emple: Solve 4{3[7 y 3 9] y 9} 5 y 3. : 4{3[7 y 3 9] y 9} 5 y 3 4{3[7 y 9] y 8} 5y 5 3 4{3[7 y ] y 8} 5y 8 4{y 36 y 8} 5y 8 4{3y 54} 5y 8 9y 6 5y 8 9y 7 5y 8 : 9y 7 5y 8 9y 5y 7 5y 5y 8 87y y y 9 3 :87y 9 9 y : Check the fil swe y i the oigil equtio 87 4{3[7 y 3 9] y 9} 5 y 3. 5

26 .. Lws of Iequlity Let A > B e lgeic iequlity d C e y qutity.... Additio: A C > B C... Sutctio: A C > B C C > A C > B C..3. Multiplictio: C < A C < B C A B C > > C C..4. Divisio: A B C < < C C..5. Recipocl: < povided A, B A B Simil lws hold fo A < B, A B, d A B. Whe multiplyig o dividig y egtive C, oe must evese the diectio of the oigil iequlity sig. Replcig ech side of the iequlity with its ecipocl lso eveses the diectio of the oigil iequlity...6. Lie Iequlity Solutio Pocess Stt with the geel fom L > R whee L d R e s descied i... Follow the sme fou-step pocess s tht give i.. modifyig pe the checks elow. Revese the diectio of the iequlity sig whe multiplyig o dividig oth sides of the iequlity y egtive qutity. Revese the diectio of the iequlity sig whe eplcig ech side of iequlity with its ecipocl. The fil swe will hve oe the fou foms >,, <, d. Oe must ememe tht i of the fou cses, hs ifiitely my solutios s opposed to oe solutio fo the lie equtio. 6

27 .. Ode of Opetios Step : Pefom ll powe isigs i the ode they occu fom left to ight Step : Pefom ll multiplictios d divisios i the ode they occu fom left to ight Step 3: Pefom ll dditios d sutctios i the ode they occu fom left to ight Step 4: If petheses e peset, fist pefom steps though 3 o s-eeded sis withi the iemost set of petheses util sigle ume is chieved. The pefom steps though 3 gi, o s-eeded sis fo the et level of petheses util ll petheses hve ee systemticlly emoved. Step 5: If fctio is peset, simulteously pefom steps though 4 fo the umeto d deomito, tetig ech s totlly-septe polem util sigle ume is chieved. Oce sigle umes hve ee chieved fo oth the umeto d the deomito, the fil divisio c e pefomed..3. Thee Meigs of Equls. Equls is the mthemticl equivlet of the Eglish ve is, the fudmetl ve of eig. A simple ut sutle use of equls i this fshio is.. Equls implies equivlecy of mig i tht the sme udelyig qutity is eig med i two diffeet wys. This c e illustted y the epessio 3 MMIII. Hee, the two divese symols o oth sides of the equls sig efe to the sme d ect udelyig qutity. 3. Equls sttes the poduct eithe itemedite o fil tht esults fom pocess o ctio. Fo emple, i the epessio 4, we e ddig two umes o the lefthd side of the equls sig. Hee, dditio c e viewed s pocess o ctio etwee the umes d. The esult o poduct fom this pocess o ctio is the sigle ume 4, which ppes o the ight-hd side of the equls sig. 7

28 .4. The Seve Petheses Rules.4.. Cosecutive pocessig sigs,,, e septed y petheses..4.. Thee o moe cosecutive pocessig sigs e septed y ested pethesis whee the ightmost sig will e i the iemost set of petheses Nested petheses e typiclly witte usig the vious cketig symols to fcilitte edig The ightmost pocessig sig d the ume to the immedite ight of the ightmost sig e oth eclosed withi the sme set of petheses Petheses my eclose siged o usiged ume y itself ut eve sig y itself Moe th oe ume c e witte iside set of petheses depedig o wht pt of the ovell pocess is emphsized Whe petheses septe umes with o iteveig multiplictio sig, multiplictio is udestood. The sme is tue if just oe plus o mius sig septes the two umes d the petheses eclose oth the ightmost ume d the septig sig Demosttig the Seve Bsic Petheses Rules 5 : Popely witte s 5..4., : Popely witte s 5..4., : Popely witte s 5 [ ]..4. thu 4 5 : Icoect pe : Coect pe.4.,.4.4, : Does ot eed petheses to chieve septio sice the 5 seves the sme pupose. Ay use of petheses would e optiol 5 : The optiol petheses, though ot eeded, emphsize the egtive5 pe : The optiol petheses emphsize the fct tht the fil outcome is egtive pe.4.5,.4.6 8

29 4 : The mdtoy petheses idicte tht 4 is multiplyig. Without the petheses, the epessio would e popely ed s the sigle ume 4, : The mdtoy petheses idicte tht 7 is multiplyig 5. Without the iteveig petheses, the epessio is popely ed s the diffeece 7 5, : The mdtoy petheses idicte tht 3 is multiplyig 5. The epessio 3 5 lso sigifies the sme, Demosttio of Use of Ode-of-Opetios with Petheses Rules to Reduce Rtiol Epessio { 8} { 8} [8 8]

30 .5. Rules fo Logithms.5.. Defiitio of Logithm to Bse > : y log if d oly if y.5.. Logithm of the Sme Bse: log.5.3. Logithm of Oe: log.5.4. Logithm of the Bse to Powe: log p p log p.5.5. Bse to the Logithm: p.5.6. Nottio fo Logithm Bse : Log log.5.7. Nottio fo Logithm Bse e : l log log N.5.8. Chge of Bse Fomul: log N log.5.9. Poduct: log MN log N log M M.5.. Quotiet: log log M log N N p.5.. Powe: log N plog N.5.. Logithmic Simplifictio Pocess m A B Let X, the p C m A B log X log p C log X log log X log log X log m p A B log C m p A log B log C A mlog B p log C Note: The use of logithms tsfoms comple lgeic epessios whee poducts ecome sums, quotiets ecome diffeeces, d epoets ecome coefficiets, mkig the mipultio of these epessios esie i some istces. e 3

31 .6. Comple Numes.6.. Defiitio of the imgiy uit i : i is defied to e the solutio to the equtio..6.. Popeties of the imgiy uit i : i i i.6.3. Defiitio of Comple Nume: Numes of the fom i whee, e el umes.6.4. Defiitio of Comple Cojugte: i i.6.5. Defiitio of Comple Modulus: i.6.6. Additio: i c di c d i.6.7. Sutctio: i c di c d i.6.8. Pocess of Comple Nume Multiplictio i c di c d c i di c d c i d c d d c i.6.9. Pocess of Comple Nume Divisio i c di i c di c di c di i c di c di c di c d c d i c d c d c d i c d c d 3

32 .7. Wht is Fuctio? The mthemticl cocept clled fuctio is foudtiol to the study of highe mthemtics. With this sttemet i mid, let us defie i wokig sese the wod fuctio: A fuctio is y pocess whee umeicl iput is tsfomed ito umeicl output with the opetig estictio tht ech uique iput must led to oe d oly oe output. Iput Side Fuctio Nme f Pocessig Rule f Output Side The ove figue is digm of the geel fuctio pocess fo fuctio med f. Fuctio mes e usully lowe-cse lettes, f, g, h, etc. Whe mthemtici sys, let f e fuctio, the etie iput-output pocess stt to fiish comes ito discussio. If two diffeet fuctio mes e eig used i oe discussio, the two diffeet fuctios e eig discussed, ofte i tems of thei eltioship to ech othe. The vile is the idepedet o iput vile; it is idepedet ecuse y specific iput vlue c e feely chose. Oce specific iput vlue is chose, the fuctio the pocesses the iput vlue vi the pocessig ule i ode to cete the output vile f, lso clled the depedet vile sice the vlue of f is etiely detemied y the ctio of the pocessig ule upo. Notice tht the comple symol f eifoces the fct tht output vlues e ceted y diect ctio of the fuctio pocess f upo the idepedet vile. Sometimes, simple y will e used to epeset the output vile f whe it is well udestood tht fuctio pocess is ideed i plce. Two moe defiitios e oted. The set of ll possile iput vlues fo fuctio f is clled the domi d is deoted y the symol Df. The set of ll possile output vlues is clled the ge d is deoted y Rf. 3

33 .8. Fuctio Alge Let f d g e fuctios, d let f e the ivese fo f.8.. Ivese Popety: f f f [ ] [ f ].8.. Additio/Sutctio: f ± g f ± g.8.3. Multiplictio: f g f g f f g g.8.4. Divisio: ; g g f f f o g f [ g ].8.5. Compositio: g o f g[ f ].8.6. Pocess fo Costuctig Ivese Fuctios Step : Stt with f f, the pocess equlity tht must e i plce fo ivese fuctio to eist. Step : Replce f with y to fom the equlity y. f Step 3: Solve fo y i tems of. The esultig y is Step 4: Veify y the popety f f f f Demosttio of.8.6: Fid f fo f 3. : f f : y 3 : y y 4 : f : f f f f f. 33

34 .9. Qudtic Equtios & Fuctios.9.. Defiitio d Discussio A complete qudtic equtio i stdd fom edy-to-esolved is equtio hvig the lgeic stuctue c whee,, c. If eithe o c, the qudtic equtio is clled icomplete. If, the qudtic equtio educes to lie equtio. All qudtic equtios hve ectly two solutios if comple solutios e llowed. Solutios e otied y eithe fctoig o y use of the qudtic fomul. If, withi the cotet of pticul polem comple solutios e ot dmissile, qudtic equtios c hve up to two el solutios. As with ll el-wold pplictios, the ume of dmissile solutios depeds o cotet..9.. Qudtic Fomul with Developmet: c c c 4 4 ± ± 4c 4 4c 4c.9.3. Solutio of Qudtic Equtios y Fomul To solve qudtic equtio usig the qudtic fomul the moe poweful of two commo methods fo solvig qudtic equtios pply the followig fou steps. Step : Rewite the qudtic equtio so it mtches the stdd fom c. 34

35 Step : Idetify the two coefficiets d costt tem,,& c. Step 3: Apply the fomul d solve. Step 4: Check you swes i the oigil equtio Solutio Discimito: 4c 4c > two el solutios 4c oe el solutio of multiplicity two 4c <.9.5. Solutio whe & : c c two comple cojugtes solutios.9.6. Solutio of Qudtic Equtios y Fctoig To solve qudtic equtio usig the fctoig method, pply the followig fou steps. Step : Rewite the qudtic equtio i stdd fom Step : Fcto the left-hd side ito two lie fctos usig the qudtic tiomil fctoig pocess.9.4. Step 3: Set ech lie fcto equl to zeo d solve. Step 4: Check swes i the oigil equtio Note: Use the qudtic fomul whe qudtic equtio cot e fctoed o is hd to fcto Qudtic-i-Fom Equtio: U U c whee U is lgeic epessio of vyig compleity Defiitio of Qudtic Fuctio: 4c f c Ais of Symmety fo Qudtic Fuctio: 4c.9.. Vete fo Qudtic Fuctio:, 4 35

36 .. Cdo s Cuic Solutio d 3 Let c e cuic equtio witte i stdd fom with Step : Set y. Afte this sustitutio, the ove cuic 3 3 c ecomes y py q whee p d 3 c q d Step : Defie u & v such tht y u v d p 3uv 3 Step 3: Sustitute fo y & p i the equtio y py q p This leds to u qu, which is qudtici-fom i 3 u. 7 Step 4: Use the qudtic fomul.9.3 to solve fo u q 4 3 q 3 7 p 3 u Step 5: Solve fo u 3 u & v whee q q 4 7 p v to oti 3u p 3 & v 3 q q 4 7 p 3 Step 6: Solve fo whee y 3 u v 3 36

37 .. Theoy of Polyomil Equtios... Let P e polyomil witte i stdd fom. The Eight Bsic Theoems... Fudmetl Theoem of Alge: Evey polyomil P of degee N hs t lest oe solutio fo which P. This solutio my e el o comple i.e. hs the fom i.... Numes Theoem fo Roots d Tuig Poits: If P is polyomil of degee N, the the equtio P hs up to N el solutios o oots. The equtio P hs ectly N oots if oe couts comple solutios of the fom i. Lstly, the gph of P will hve up to N tuig poits which icludes oth eltive mim d miim...3. Rel Root Theoem: If P is of odd degee hvig ll el coefficiets, the P hs t lest oe el oot...4. Rtiol Root Theoem: If P hs ll itege coefficiets, the y tiol oots fo the equtio P p must hve the fom q whee p is fcto of the costt coefficiet d q is fcto of the led coefficiet. Note: This esult is used to fom tiol-oot possiility list...5. Comple Cojugte Pi Root Theoem: Suppose P hs ll el coefficiets. If i is oot fo P with P i, the P i...6. Itiol Sud Pi Root Theoem: Suppose P hs ll tiol coefficiets. If is oot fo P with P, the P. 37

38 ..7. Remide Theoem: If P is divided y c, the the emide R is equl to P c. Note: this esult is etesively used to evlute give polyomil P t vious vlues of...8. Fcto Theoem: If c is y ume with P c, the c is fcto of P. This mes P c Q whee Q is ew, educed polyomil hvig degee oe less th P. The covese P c Q P c is lso tue. The Fou Advced Theoems..9. Root Loctio Theoem: Let, e itevl o the is with P P <. The thee is vlue, such tht P.... Root Boudig Theoem: Divide P y d to oti P d Q R. Cse d > : If oth R d ll the coefficiets of Q e positive, the P hs o oot > d. Cse d < : If the oots of Q ltete i sig with the emide R i syc t the ed the P hs o oot < d. Note: Coefficiets of zeo c e couted eithe s positive o egtive which eve wy helps i the susequet detemitio.... Desctes Rule of Sigs: Age P i stdd ode s show i the title. The ume of positive el solutios equls the ume of coefficiet sig vitios o tht ume decesed y eve ume. Likewise, the ume of egtive el solutios equls the ume of coefficiet sig vitios i o tht ume decesed y eve ume. P... Tuig Poit Theoem: Let polyomil P hve degee N. The the ume of tuig poits fo polyomil P c ot eceed N. 38

39 .. Detemits d Cme s Rule... Two y Two Detemit Epsio: d c c d... Thee y Thee Detemit Epsio: c e f d f d e d e f c h i g i g h g h i ei fh di fg c dh eg ei hf fg di cdh ceg..3. Cme s Rule fo Two-y-Two Lie System Give y e c dy f with D c d The e f d d D y e c f D..4. Cme s Rule fo Thee-y-Thee Lie System Give y cz j c d ey fz k with D d e f g hy iz l g h i The j c j c k e f d k f l h i g l i, y, z D D d g e h D j k l 39

40 Di..5. Solutio Types i i D D i, D i D i, D i hs ifiite solutios D i, D i hs uique solutio D i, D i hs o solutio.3. Biomil Theoem Let d e positive iteges with..3.. Defiitio of!:!...,.3.. Specil Fctoils:! d!!.3.3. Comitoil Symol:!!.3.4. Summtio Symols:... i 3 4 i i k k k k 3... i k.3.5. Biomil Theoem: i i i.3.6. Sum of Biomil Coefficiets whe : i i i i.3.7. Fomul fo the th Tem: i 4

41 :.3.8. Pscl s Tigle fo Aithmetic Seies.4.. Defiitio: S i i whee is the commo icemet.4.. Summtio Fomul fo S : S [ ].5. Geometic Seies i.5.. Defiitio: G whee is the commo tio i.5.. Summtio Fomul fo G : G i G G i G i G i i i i i i.5.3. Ifiite Sum Povided < < : i i i 4

42 .6. Boole Alge I the followig tles, the popositios o Flse F..6.. Elemety Tuth Tle: p & q e eithe Tue T d : o : egtio ~: implies, p q ~ p ~ q p q p q p q p q T T F F T T T T T F F T F T F F F T T F F T F F F F T T T F T T.6.. Tuth Tle fo the Eclusive O p q p e q T T F T F T F T T F F F e :.6.3. Modus Poes: Let p q & p T. The, q T Chi Rule: Let p q & q. The p T Modus Tolles: Let p q & q F. The ~ q ~ p T Fllcy of Affimig the Cosequet: Let p q & q T.The q p F Fllcy of Deyig the Atecedet: Let p q & p F. The ~ p ~ q F Disjuctive Syllogism fo the Eclusive O: p Let p e q T & q F. The T 4

43 .6.9. Demosttio tht the Eglish doule-egtive i the slg epessio I do t got oe ctully ffims the opposite of wht is iteded. Step Phse Commet I do ot hve y The oigil popositio p s iteded I do hve oe Assume p T I do ot hve oe Negtio of p : ~ p F 3 I do t hve oe Pope cotcted fom of 3: ~ p F 4 I do t got oe Slg vesio of 3 5 I hve some Logicl cosequece of 3: ~ p F ~ ~ p T.7. Vitio o Popotiolity Fomuls.7.. Diect: y k k.7.. Ivese: y.7.3. Joit: z ky k.7.4. Ivese Joit: z y.7.5. Diect to Powe: y k k.7.6. Ivese to Powe: y 43

44 . Geomety.. The Pllel Postultes... Let poit eside outside give lie. The thee is ectly oe lie pssig though the poit pllel to the give lie.... Let poit eside outside give lie. The thee is ectly oe lie pssig though the poit pepedicul to the give lie...3. Two lies oth pllel to thid lie e pllel to ech othe...4. If tsvese lie itesects two pllel lies, the coespodig gles i the figues so fomed e coguet...5. If tsvese lie itesects two lies d mkes coguet, coespodig gles i the figues so fomed, the the two oigil lies e pllel... Agles d Lies α β α β α β 8 α β 9... Complimety Agles: Two gles α, β with β 9 α. 44

45 ... Supplemety Agles: Two gles α, β with α β Lie Sum of Agles: The sum of the two gles α, β fomed whe stight lie is itesected y lie segmet is equl to Acute Agle: A gle less th Right Agle: A gle ectly equl to Otuse Agle: A gle gete th 9.3. Tigles γ α β γ 8 α c β.3.. Tigul Sum of Agles: The sum of the thee iteio gles α, β, γ i y tigle is equl to Acute Tigle: A tigle whee ll thee iteio glesα, β, γ e cute.3.3. Right Tigle: A tigle whee oe iteio gle fom the tid α, β, γ is equl to Otuse Tigle: A tigle whee oe iteio gle fom the tid α, β, γ is gete th Sclee Tigle: A tigle whee o two of the thee side-legths,, c e equl to othe.3.6. Isosceles Tigle: A tigle whee ectly two of the side-legths,, c e equl to ech othe.3.7. Equiltel Tigle: A tigle whee ll thee sidelegths,, c e ideticl c o ll thee gles α, β, γ e equl with α β γ 6 45

46 .3.8. Coguet Tigles: Two tigles e coguet equl if they hve ideticl iteio gles d side-legths Simil Tigles: Two tigles e simil if they hve ideticl iteio gles..3.. Icluded Agle: The gle tht is etwee two give sides.3.. Opposite Agle: The gle opposite give side.3.. Icluded Side: The side tht is etwee two give gles.3.3. Opposite Side: The side opposite give gle.4. Coguet Tigles Give the coguet two tigles s show elow γ e ω d α c β φ.4.. Side-Agle-Side SAS: If y two side-legths d the icluded gle e ideticl, the the two tigles e coguet. Emple: & α & c e & φ & f.4.. Agle-Side-Agle ASA: If y two gles d the icluded side e ideticl, the the two tigles e coguet. Emple: α & c & β φ & f & ϕ.4.3. Side-Side-Side SSS: If the thee side-legths e ideticl, the the tigles e coguet. Emple: & c & e & f & d.4.4. Thee Attiutes Ideticl: If y thee ttiutes side-legths d gles e equl with t lest oe ttiute eig side-legth, the the two tigles e coguet. These othe cses e of the fom Agle-Agle-Side AAS o Side-Side-Agle SSA. Emple SSA: & & β e & d & ϕ Emple AAS: & & & & d f α β φ ϕ ϕ 46

47 .5. Simil Tigles Give the two simil tigles s show elow ω γ e d α β c φ ϕ f.5.. Miiml Coditio fo Simility: If y two gles e ideticl AA, the the tigles e simil. Suppose The α φ & β ϕ α β γ 8 α 8 β γ 8 & φ ϕ ϖ 8 ϕ ϖ φ.5.. Rtio lws fo Simil Tigles: Give simil c tigles s show ove, the e f d.6. Pl Figues A is the pl e, P is the peimete, is the ume of sides..6.. Degee Sum of Iteio Agles i Geel Polygo: D 8 [ ] D 54 6 D Sque: A s : P 4s, s is the legth of side s 47

48 .6.3. Rectgle: A h : P h, & h e the se d height h.6.4. Tigle: A h, & h e the se d ltitude h.6.5. Pllelogm: A h, & h e the se d ltitude h.6.6. Tpezoid: A B h, B & e the two pllel ses d h is the ltitude h B π π.6.7. Cicle: A : P whee is the dius, o P πd whee d, the dimete Ellipse: A π ; & e the hlf legths of the mjo & mio es 48

49 .7. Solid Figues A is totl sufce e, V is the volume.7.. Cue: A 6s V s 3 :, s is the legth of side s π Sphee: A 4 : V, is the dius 3 π.7.3. Cylide: A π πl : V π l, & l the dius d legth l e.7.4. Coe: A π πt : V π h, & t & h e the dius, slt height, d ltitude 3 h t 49

50 .7.5. Pymid sque se: A s st : V 3 s h, s & t & h e the side, slt height, d ltitude s h t.8. Pythgoe Theoem.8.. Sttemet: Let ight tigle ABC hve oe side AC of legth, secod side AB of legth y, d hypoteuse log side BC of legth z. The z y B y A z C.8.. Tditiol Algeic Poof: Costuct ig sque y igig togethe fou coguet ight tigles. z y 5

51 The e of the ig sque is give y A y A z 4. Equtig: y y z 4 y y z y. y z z y y, o equivletly y.8.3. Visul Pe-Algeic Pythgoe Poof: The ide is to oseve tht the two five-sided iegul polygos o eithe side of the dotted lie hve equivlet es. Tkig wy thee coguet ight tigles fom ech e leds to the desied Pythgoe equlity Pythgoe Tiples: Positive iteges, such tht L M N L M, N.8.5. Geetig Fomuls fo Pythgoe tiples: Let m, with m > > e iteges. The M m, N m, d L m 5

52 5.9. Heo s Fomul Let c s e the semi-peimete of geel tigle d A e the itel e eclosed y the sme..9.. Heo s Fomul: c s s s s A.9.. Deivtio Usig Pythgoe Theoem: : Cete two equtios fo the ukows h d : : c h E h E : Sutct E fom E d solve fo c c c c c : 3 Sustitute the vlue fo ito E c c h : 4 Solve fo h [ ] 4 4 c c c h c h c

53 53 [ ] { } [ ] { } [ ] { }[ ] { } { }{ }{ }{ } c c c c c h c c c h c c c c c h : 5 Solve fo e usig ch A. { }{ }{ }{ } { }{ }{ }{ } 6 4 c c c c A c c c c c c A : 6 Sustitute c s d simplify. } { c s s s s A s s s c s A s s s c s A.. Golde Rtio... Defiitio: Let p e the semi-peimete of ectgle whose se d height e i the popotio show, defiig the Golde Rtioφ. Solvig fo leds to 68. φ. φ

54 ... Golde Tigles: Tigles whose sides e popotioed to the Golde Rtio. Two emples e show elow. B B 36 φ 8 C 7 A C A D.. Distce d Lie Fomuls Let, y d, y e two poits whee >.... -D Distce Fomul: D y y... 3-D Distce Fomul: Fo the poits, y, z d, y,, z D y y z z y y..3. Midpoit Fomul:, Lie Fomuls y y..4. Slope of Lie: m..5. Poit/Slope Fom: y y m..6. Geel Fom: A By C..7. Slope/Itecept Fom: y m whee, m d, e the d y Itecepts: 54

55 y..8. Itecept/Itecept Fom: whee, d, e the d y itecepts..9. Slope Reltioship etwee two Pllel Lies L d L hvig slopes m d m : m m... Slope Reltioship etwee two Pepedicul Lies L d L hvig slopes m d m : m m... Slope of Lie Pepedicul to Lie of Slope m : m.. Fomuls fo Coic Sectios... Geel: A By Cy D Ey F... Cicle of Rdius Ceteed t h, k : h y k..3. Ellipse Ceteed t h y k h, k : I If >, the two foci e o the lie y k d e give y h c, k & h c, k whee. II If >, the two foci e o the lie h d e give y h, k c & h, k c whee c...4. Hypeol Ceteed t h, k : h y k y k h o h I Whe is to the left of the mius sig, the two foci e o the lie y k d e give y h c, k & h c, k whee c. c 55

56 y k II Whe is to the left of the mius sig, the two foci e o the lie h d e give y h, k c & h, k c whee c...5. Pol with Vete t h, k d Focl Legth p : y k 4 p h o h 4 p y k I Fo y k, the focus is h p, k d the diecti is give y the lie h p. II Fo h, the focus is h, k p d the diecti is give y the lie. y k p..6. Tsfomtio Pocess fo Removl of y Tem i the Geel Coic Equtio A By Cy D Ey F : B Step : Set t θ d solve fo θ. A C Step : let cosθ y siθ y siθ y cosθ Step 3: Sustitute the vlues fo, y otied i Step ito A By Cy D Ey F. Step 4: Reduce. The fil esult should e of the fom A C y D E y F. 56

57 3. Tigoomety 3.. Bsic Defiitios of Tigoometic Fuctios & Tigoometic Ivese Fuctios z y α Let the figue ove e ight tigle with oe side of legth, secod side of legth y, d hypoteuse of legth z. The gle α is opposite the side of legth. The si tigoometic fuctios whee ech is fuctio of α e defied s follows: Z : Aity Z is Ivese whe Z is y 3... siα siα y si y α z 3... cosα cosα cos α z y y y tα tα t α cotα cotα cot α y y y z secα sec α sec α z cscα csc α csc α y y y si Note: is lso kow s csi. Likewise, the othe iveses e lso kow s ccos, ct, c cot, csec d c csc. 57

58 3.. Fudmetl Defiitio-Bsed Idetities 3... csc α si α 3... sec α cos α si α t α cos α cos α cot α si α t α cot α 3.3. Pythgoe Idetities si α cos α t α sec α cot α csc α 3.4. Negtive Agle Idetities si α si α cos α cos α t α t α cot α cot α 3.5. Sum d Diffeece Idetities α si β si α cos β cos αsi β si α β si α cos β cos αsi β cos α β cos α cos β si αsi β cos β cos α cos β si αsi β α 58

59 t α t β t α β t α t β t α t β t α β t α t β Deivtio of Fomuls fo cos β si α β : α d I the figue elow, ech coodite of the poit {cos α β,si α β } is decomposed ito two compoets usig oth defiitios fo the sie d cosie i 3... d 3... y {cos α β,si α β } si β si α siβ cosβ α y si β cos α {cos α,si α} y si αcos β β, α, cos αcos β Fom the figue, we hve cos α β cos α β cos αcos β si αsi β si α β y y. si α β si αcos β si β cos α 59

60 3.6. Doule Agle Idetities si α si α cos α α α α cos cos si α cos cos si t α t α t α α 3.7. Hlf Agle Idetities α α cos α si ± α cos α cos ± α cos α t ± cos α si α cos α cos α si α 3.8. Geel Tigle Fomuls Applicle to ll tigles: ight d o-ight z β y α θ Sum of Iteio Agles: α β θ 8 lso.3.. si α si β si θ Lw of Sies: y z 6

61 Lw of Cosies: y z z cos α y z yz cos β c z y y cos θ Ae Fomuls fo Geel Tigle: A z si α A yz si β c A y si θ Deivtio of Lw of Sies d Cosies: Let ABC e geel tigle d dop pepedicul fom the pe s show. C A β γ h y c y α B Fo the Lw of Sies we hve h : si α h si α h : si β h si β 3 : si α si β si β si α The lst equlity is esily eteded to iclude the thid gle γ. 6

62 Fo the Lw of Cosies we hve h siα. : Solve fo y d i tems of the gle α y cos α y cos α c y c cos α : Use the Pythgoe Theoem to complete the deivtio. h [ c cos α] c c ccos α ccos α c [ si α] ccos α cos α si α Simil epessios c e witte fo the emiig two sides Ac d Secto Fomuls θ s Ac Legth s : s θ Ae of Secto: A θ 3.. Degee/Rdi Reltioship 3... Bsic Covesio: 8 π dis 6

63 3... Covesio Fomuls: Fom To Multiply y Rdis Degees Degees Rdis 8 π π Additio of Sie d Cosie siθ cosθ k si θ α whee k α si o α cos 3.. Pol Fom of Comple Numes 3... i cos i si whee θ, θ T iθ θ 3... Defiitio of e : e i cosθ i siθ iπ Eule s Fmous Equlity: e iθ iθ De-Moive s Theoem: e e o [ cosθ i siθ ] cos[ θ ] i si[ θ ] θ 63

64 3..5. Pol Fom Multiplictio: iα iβ i α β e e e Pol Fom Divisio: e iα i α β e iβ e 3.3. Rectgul to Pol Coodites, y, θ cosθ, y siθ y, θ t y / 3.4. Tigoometic Vlues fom Right Tigles I the ight tigle elow, let si α si α. α The cos α t α cot α sec α csc α 64

65 4. Elemety Vecto Alge 4.. Bsic Defiitios d Popeties Let V v, v,, U u, u, e two vectos. v3 u3 U θ V 4... Sum d/o Diffeece: U ± V U ± V u ± v, u ± v, u3 ± v Scl Multiplictio: α U αu, αu, αu Negtive Vecto: U U Zeo Vecto:,, Vecto Legth: U u u u Uit Vecto Pllel tov : V V Two Pllel Vectos: V U mes thee is scl c such tht V c U 4.. Dot Poducts 4... Defiitio of Dot Poduct: U V uv u v u3v3 U V 4... Agleθ Betwee Two Vectos: cosθ U V Othogol Vectos: U V 65

66 Pojectio of U otov : [ ] cos V V U V V V V U V V V U U poj V θ 4.3. Coss Poducts Defiitio of Coss Poduct: 3 3 v v v u u u k j i V U Oiettio of V U ; Othogol to BothU dv : V U V V U U Ae of Pllelogm: θ si V U V U A Itepettio of the Tiple Scl Poduct: w w w v v v u u u W V U The tiple scl poduct is umeiclly equl to the volume of the pllelepiped t the top of the et pge U V θ V U

67 U W V 4.4. Lie d Ple Equtios Give poit P,, c Lie Pllel to P Pssig Though, y, z : If, y, z is poit o the lie, the y y z z c Ple Noml to P Pssig Though, y, z. If, y, z is poit o the ple, the,, c, y y, z z Distce D etwee poit & ple: If poit is give y, y, z d y cz d is ple, the D y cz c d 4.5. Miscelleous Vecto Equtios The Thee Diectio Cosies: v v v3 cosα,cos β,cosγ V V V Defiitio of Wok: costt foce F log the pth PQ : W F PQ poj F PQ PQ 67

68 5. Elemety Clculus 5.. Wht is Limit? Limits e foudtiol to clculus d will lwys e so. Limits led to esults uotile y lge loe. So wht is limit? A limit is umeicl tget, tget cquied d locked. Coside the epessio 7 whee is idepedet vile. The ow poits to tget o the ight, i this cse the ume 7. The vile o the left is tgetig 7 i mode smt-wepo sese. This mes is movig, movig towds tget, closig ge, d pogmmed to mege evetully with the tget. Notice tht the qutity is tue idepedet vile i tht hs ee luched d set i motio towds tget, tget tht cot escpe fom its sights. Idepedet viles usully fid themselves emedded iside lgeic o tscedetl epessio of some sot, which is eig used s pocessig ule fo fuctio. Coside the epessio 3 whee the idepedet vile is out to e set o the missio 5. Does the etie epessio 3 i tu tget umeicl vlue s 5? A wy to phse this questio usig ew type of mthemticl ottio might e t g et 3? Itepetig the ottio, we e 5 skig if the dymic output stem fom the epessio 3 tgets umeicl vlue i the mode smt-wepo sese s the eqully-dymic tgets the vlue 5. Mthemticl judgmet sys yes; the output stem tgets the vlue 7. Hece, we complete ou ew ottio s t get This epltio is esole ecept fo oe little polem: the wod tget is owhee to e foud i clculus tets. The tditiol eplcemet weighig i with 3 yes of histoy is the wod limit, which leds to the followig wokig defiitio: Wokig Defiitio: A limit is tget i the mode smt-wepo sese. I the ove emple, we will wite lim

69 5.. Wht is Diffeetil? The diffeetil cocept is oe of the two coe cocepts udelyig clculus, limits eig the othe. Wee is Scottish wod tht mes vey smll, tiy, dimiutive, o miuscule. I the cotet of clculus, wee c e used i simil fshio to help epli the cocept of diffeetil, lso clled ifiitesiml. To hve diffeetil, we fist must hve vile,, y, z etc. Oce we hve vile, sy, we c cete secody qutity d, which is clled the diffeetil of the vile. Wht ectly is this d, ed dee? The qutity d is vey smll, tiy, dimiutive, o miuscule umeicl mout whe comped to the oigil. Moeove, it is the vey smll size of d tht mkes it, y defiitio, wee. How smll? I mthemticl tems, the followig two coditios hold: d < d << d < <<. The two ove coditios stte d is smll eough to gutee tht oth its poduct d quotiet with the oigil qutity is still vey smll d much, much close to zeo th to oe the meig of the symol <<. Both iequlities imply tht d is lso vey smll whe cosideed idepedetly < d <<. Lstly, oth iequlities stte tht d >, which igs us to the followig vey impott poit: lthough vey smll, the qutity d is eve zeo. Oe c lso thik of d s the fil h i limit pocess lim whee the pocess uptly stops just shot of tget, h i effect svig the pidly vishig h fom disppeig ito olivio! Thikig of d i this fshio mkes the diffeetil pepckged o foze limit of sots. Diffeetils e desiged to e so smll tht secod-ode d highe tems ivolvig diffeetils, such s 7 d, c e totlly igoed i ssocited lgeic epessios. This fil popety distiguishes the diffeetil s topic elogig to the suject of clculus. 69

70 5.3. Bsic Diffeetitio Rules Limit Defiitio of Deivtive: f h f f ' lim h h Diffeetitio Pocess Idicto: [] Costt: [ k ] Powe:[ ], c e y epoet Coefficiet:[ f ] f ' Sum/Diffeece:[ f ± g ] f ± g Poduct:[ f g ] f g' g f ' f g f ' f g' Quotiet: g g Chi:[ f g ] f ' g g' Ivese: [ f ] f ' f Geelized Powe: [{ f } ] { f } f ' ; Agi, c e y epoet 5.4. Tscedetl Diffeetitio [l ] [log ] l [ e ] e [ ] l [si ] cos 7

71 [si ] [cos ] si [cos ] [t ] sec [t ] [sec ] sec t [sec ] 5.5. Bsic Atidiffeetitio Rules Atidiffeetitio Pocess Idicto: Costt: kd k C Coefficiet: f d f d Powe Rule fo : d C Powe Rule fo : d d l C Sum: [ g ] d f d f g d Diffeece: f g d f d g d [ ] Pts: f g d f g g f d Chi: f g g d f g C 7

72 5.5.. Geelized Powe Rule fo : [ f ] f d [ f ] C Geelized Powe Rule fo : f d l f C, f f f Geel Epoetil: e f d e C 5.6. Tscedetl Atidiffeetitio l d l C e d e C e d e C d C l cos d si C si d cos C t d l cos C cot d l si C sec d l sec t C sec t d sec C sec d t C csc d l csc cot C csc d cot C 7

73 d si C d t C d l C 5.7. Lies d Appoimtio Tget Lie t, f : y f f Noml Lie t, f : y f f Lie Appoimtio: f f f Secod Ode Appoimtio: f f f f f Newto s Itetive Fomul: f Diffeetil Equlities: y f dy f d f d f f d F d F f d 5.8. Itepettio of Defiite Itegl At lest thee itepettios e vlid fo the defiite itegl. Fist Itepettio: As pocessig symol fo fuctios, the defiite itegl f d istucts the opeto to stt the pocess y fidig F the pimy tideivtive fo f d d fiish it y evlutig the qutity F F F. This itepettio is pue pocess-to-poduct with o cotet. 73

74 Secod Itepettio: As summtio symol fo diffeetil qutities, f d sigls to the opeto tht myids of ifiitesiml qutities of the fom f d e eig cotiuously summed o the itevl [, ] with the summtio pocess sttig t d edig t. Depedig o the cotet fo give polem, such s summig e ude cuve, the diffeetil qutities f d d susequet totl c tke o viety of meigs. This mkes cotiuous summig poweful tool fo solvig el-wold polems. The fct tht cotiuous sums c lso e evluted y f d F F F is key cosequece of the Fudmetl Theoem of Clculus Thid Itepettio: The defiite itegl f d c e itepeted s poit solutio y to y eplicit diffeetil equtio hvig the geel fom dy f d : y. I this itepettio f d is fist modified y itegtig ove the vile suitevl [, z] [, ]. This leds to z y z f d F z F. Sustitutig gives the stted oudy coditio y F F d sustitutig gives y F F f d. I this cotet, the fuctio y z F z F, s uique solutio to dy f d : y, c lso e itepeted s cotiuous uig sum fom to z. 74

75 5.9. The Fudmetl Theoem of Clculus Let f d e defiite itegl epesetig cotiuous summtio pocess, d let F e such tht F f. The, f d c e evluted y the ltetive pocess f d F F F. Note: A cotiuous summtio o dditio pocess o the itevl[, ] sums millios upo millios of cosecutive, tiy qutities fom to whee ech idividul qutity hs the geel fom f d. 5.. Geometic Itegl Fomuls 5... Ae Betwee two Cuves fo f g o [, ] : A [ f g ] d 5... Ae Ude f o [, ] : A f d Volume of Revolutio out Ais Usig Disks: V π [ f ] d Volume of Revolutio out y Ais usig Shells: V π f d Ac Legth: s [ f ] d 75

76 5..6. Revolved Sufce Ae out Ais: SA π f [ f ' ] d Revolved Sufce Ae out y Ais: SA π [ f ' ] Totl Wok with Vile Foce F o [, ] : W F d 5.. Select Odiy Diffeetil Equtios ODE dy 5... Fist Ode Lie: f y g d dy 5... Beoulli Equtio: f y g y d ODE Seple if it educes to: g y dy f d dv Fllig Body with Dg: m mg kv dt Costt Rte Gowth o Decy: dy ky : y y dt dy Logistic Gowth: k L y y : y y dt Cotiuous Piciple Gowth: dp P c : P P dt Newto s Lw i Oe Dimesio: d mv F dt Newto s Lw i Thee d Dimesios: mv F dt d 76

77 Pocess fo Solvig Lie ODE Step: Let F e such tht f F Step : Fomulte the itegtig fcto F e Step 3: Multiply oth sides of g y f d dy y F e g e ye d d g e y f e d dy e F F F F F Step 4: Pefom the idefiite itegtio. [ ] F F F F F Ce d g e e y y C d g e y e 5.. Lplce Tsfom; Geel Popeties 5... Defiitio: ] [ s F dt e t f t f L st 5... Lie Opeto Popety: ] [ s G s F t g t f L Tsfom of the Deivtive:... ] [ f f s f s s F s t f L Deivtive of the Tsfom: t f t s F Tsfom of the Defiite Itegl: s s F d f L t / ] [ τ τ

78 5..6. Tsfom of the Covolutio: t f τ g t τ dτ F s G s Fist Shiftig Theoem: e t f t F s Tsfom of Uit Step Fuctio U t whee U t o [, ] d U t s e o, ]. U t s Secod Shiftig Theoem: s f t U t e F s 5.3. Lplce Tsfom: Specific Tsfoms Eties e oe-to-oe coespodece etwee f t d F s / s t / s t!/ s e t / s te t / s t t e!/ s k si kt s k k si kt s s 4k ks t si kt s k s cos kt s k s k cos kt s s 4k 78

79 cos k s k s kt t sih k s k kt sih k s s k kt sih k s ks kt t cosh k s s kt cosh k s s k s kt cosh k s k s kt t si k s k kt e t sih k s k kt e t s s e e t t cos k s s kt e t cosh k s s kt e t s s s e e t t

80 6. Moey d Fice 6.. Wht is Iteest? Iteest ffects just out evey dult i Ameic. If you e idepedet, ow c o home o oth, o hve cedit cd o two, you poly py o hve pid iteest. So, wht ectly is iteest? Iteest is et chge fo the use of moey. As et chge fo the use of housig ccumultes ove time, likewise, iteest chge fo the use of moey lso ccumultes ove time. Iteest is omlly stted i tems of pecetge iteest te such s 8. Just s velocity is te of distce ccumultio e.g. 6, pecetge iteest te is miles hou velocity of pecet ccumultio. Whe divig i Ameic, the customy uits of velocity e miles pe hou. Likewise, the customy uits fo iteest te e pecet pe ye. The ede should e we tht othe th customy uits my e used i ceti situtios. Fo emple, i spce tvel 7 miles is used to descie escpe velocity fom plet eth; d, whe computig cedit-cd chge, mothly iteest te of.5 moth my e used. Both velocity d pecetge iteest te eed to e multiplied y time specified i mtchig uits i ode to oti the totl mout ccumulted, eithe miles o pecet, s i the two epessios D miles 75 hou hous 75 3 miles o pecet 3 moths 7 pecet. moth Oce the totl ccumulted iteest is computed, it is the multiplied y the mout oowed, clled the picipl P, i ode to oti the totl ccumulted iteest chge I The totl ccumulted iteest chge I, the picipl P, the pecetgeiteest te simply clled the iteest te, d the time t duig which fied picipl is oowed e elted y the fudmetl fomul I Pt. This sic fomul pplies s log s the picipl P d the iteest te emi costt thoughout the dutio of the ccumultio time t. ye sec 8

81 Fo the emiig susectios i 6., the followig pply. α : Aul gowth te s i the gowth te of voluty cotiutios to fud A : Totl mout gied o owed D : Peiodic deposit te weekly, mothly, o ully th D : Deposit mde t the stt of the i compoudig peiod i FV : Futue vlue i : Aul ifltio te L : Iitil Lump Sum M : Mothly pymet : Nume of compoudig peiods pe ye P : Amout iitilly oowed o deposited PV : Peset vlue : Aul iteest te eff : Effective ul iteest te SM : Totl sum of pymets t : Time peiod i yes fo ivestmet T : Time peiod i yes fo lo 6.. Simple Iteest 6... Accued Iteest: I P T 6... Totl epymet ove T : A P PT P T P T Mothly pymet ove T : M T 6.3. Compoud d Cotiuous Iteest t Compouded Gowth: A P t Cotiuous Gowth: A Pe Aully Compouded Ifltio Rte i : t A P i Cotiuous Aul Ifltio Rte i : A Pe Note: ifltio te c e mthemticlly teted s egtive iteest te, thus the use of the egtive sig i d it 8

82 6.4. Effective Iteest Rtes Simple Iteest: T eff T Compoud Iteest: eff Cotiuous Iteest: eff e A Give P, A, T : eff T P 6.5. Peset-to-Futue Vlue Fomuls Compoud Iteest: t FV FV PV PV Aul Compoudig with eff : t FV FV PV eff PV t eff Costt Aul Ifltio Rte with Yely Compoudig: Replce eff with i i Cotiuous Compoudig: t FV FV PVe PV t e Simple Iteest: FV FV PV t PV t 6.6. Peset Vlue of Futue Deposit Stem Coditios: compoudig peiods pe ye; totl tem t yes with t compoudig peiods; ul iteest te ; t ideticl deposits D mde t egiig of ech compoudig peiod Peiodic Deposit with o Fil Deposit D t : D t PV { } t 8

83 6.6.. Peiodic Deposit with Fil Deposit D t : D t PV { } Aul Deposit with o Fil Deposit D t : D t PV { eff eff } eff Aul Deposit with Fil Deposit D t : D t PV { } eff eff 6.7. Peset Vlue of Futue Deposit Stem Coupled with Iitil Lump Sum L > D Assume the iitil lump sum Peiodic Deposit with o Fil Deposit D t : t D t PV L D { } Peiodic Deposit with Fil Deposit D t : t D t PV L D { } Aul Deposit with o Fil Deposit D t : t D t PV L D eff eff { } Aul Deposit with Fil Deposit D t : t D t PV L D eff eff eff eff { } 6.8. Peset Vlue of Cotiuous Futue Deposit Stem D t Aul Deposit Oly: PV e eff 83

84 6.8.. Aul Deposit plus Lump t D t Sum: PV Le e t Icesig Aul Deposit De α : D t αt PV e e α plus Lump Sum: t D t αt PV Le e e α 6.9. Types of Retiemet Svigs Accouts STANDARD IRA Sposoed y Idividul Tes o cotiutios d iteest e defeed util withdw $3/ye $6/ye fo joitly filig couples Withdwls c egi t ge 59.5, must egi t 7.5 Susttil pelty fo ely withdwl Limited hei ights ROTH IRA Sposoed y Idividul Tes o cotiutios pid ow. No tes o y poceeds withdw $3/ye $6/ye fo joitly filig couples Withdwls c egi t ge 59.5 Lesse pelty fo ely withdwl Susttil hei ights 4 K Sposoed y Compy Tes o cotiutios d iteest e defeed util withdw Iceses evey ye. Cuetly $5,. Withdwls c egi t ge 59.5, must egi t 7.5 Susttil pelty fo ely withdwl Limited hei ights KEOGH PLAN Pl fo self employed Tes o cotiutios d iteest e defeed util withdw Up to 5 of icome Withdwls c egi t ge 59.5, must egi t 7.5 Susttil pelty fo ely withdwl Limited hei ights 84

85 6.. Lo Amotiztio Assume mothly pymets M 6... Fist Moth s Iteest: 6... Amout of Pymet: P I st P T [ ] M Totl Lo Repymet: SM TM Totl Iteest Pid: TM P I totl Pyoff PO j fte the j M PO j P th j Pymet: j { } Amout M Pj of M Pj M P th j Pymet to Piciple: j Amout M Ij of Iteest: M M M Ij Pj th j Pymet to Pos d Cos of Log-Tem Motgges: PROS Icesed totl motgge costs e ptilly defyed y t eks d ifltio vi pyoff y chepe dolls Allows the oowe to uy moe house sooe: with ifltio, sooe mes chepe Histoiclly, ifltio of home puchse pices cotiutes moe to home equity uildup th home equity uildup y motgge eductio CONS Totl motgge costs e much moe ove time Home equity uildup y motgge eductio is much slowe fo log-tem motgges Motgge is moe vulele to pesol misfotue such s sickess o jo loss 85

86 6.. Auity Fomuls Note: Use the lo motiztio fomuls sice uities e othig moe th los whee the oles of the istitutio d the idividul e evesed. 6.. Mkup d Mkdow C : Cost OP : Old pice NP : New pice P ; Give pecet s deciml equivlet 6... Mkup Bsed o Oigil Cost: NP P C 6... Mkup Bsed o Cost plus New Pice: C P NP NP Mkup Bsed o Old Pice: NP P OP Mkdow Bsed o Old Pice: NP P OP Pecet give Old d New Pice: P NP / OP 6.3. Clculus of Fice Geel Diffeetil Equtio of Elemety dp Fice: t P D t : P P dt Diffeetil Equtio fo Cotiuous Piciple Gowth o Cotiuous Lo Reductio Assumig Costt Iteest Rte d Fied Aul Deposits/Pymets dp P ± D : P P dt t D t P t P e ± e Peset Vlue of Totl Motgge Repymet: T T T i T P e it i P e e A PV e dt APV T T e e 86

87 87 7. Poility d Sttistics 7.. Poility Fomuls Let U e uivesl set cosistig of ll possile evets. Let Φ e the empty set cosistig of o evet. Let U B A, 7... Bsic Fomul: wys of ume totl wys of ume fvole P 7... Fudmetl Popeties: Φ P P U Ode Reltioship: A P U A Complemet Lw: ~ A P A P Additio Lw: B A P B P A P B A P Coditiol Poility Lw: A P B A P A B P B P B A P B A P Multiplictio Lw: A B P A P B A P B A P B P B A P Defiitio of Idepedet Evets IE: B Φ A IE Multiplictio Lw: B P A P B A P

88 7.. Bsic Sttisticl Defiitios 7... Set: ggegte of idividul items imte o iimte 7... Elemet: pticul item i the set Osevtio: y ttiute of iteest ssocited with the elemet Sttistic: y mesuemet of iteest ssocited with the elemet. Ay sttistic is osevtio, ut ot ll osevtios e sttistics Dt set: set whose elemets e sttistics Sttistics: the sciece of dwig coclusios fom the totlity of osevtios oth sttistics d othe ttiutes geeted fom set of iteest Popultio: the totlity of elemets tht oe wishes to study y mkig osevtios Smple: tht popultio suset tht oe hs the esouces to study Smple Sttistic: y sttistic ssocited with smple 7... Popultio Sttistic: y sttistic ssocited with popultio 7... Rdom smple: smple whee ll popultio elemets hve equl poility of ccess 7... Ifeece: the sciece of usig smple sttistics to pedict popultio sttistics Bief Discussio Usig the Aove Defiitios Let set cosist of N elemets { E, E, E3,..., EN } whee thee hs ee oseved oe sttistic of simil tue fo ech elemet. The dt set of ll oseved sttistics is deoted y {,, 3,..., N}. The coespodig k-odeed dt set is e-listig of the idividul sttistics {,, 3,..., N} i umeicl ode fom smllest to lgest. Dt sets c come fom eithe popultios o fom smples. Most dt sets will e cosideed smples. As such, the smple sttistics otied fom the smple will e utilized to mke ifeetil pedictios fo coespodig popultio sttistics chcteizig much lge popultio. Ifeece pocesses e vlid if d oly if oe c e ssued tht the smple otied is dom smple. 88

89 The digm elow suppots sectios 7. though 7.4 y illusttig some of the key cocepts. Smlle smple E, E,... E } { N Hvig kow sttistics,,... } d, s { N Did E i hve equlpoility ccess? E i Much lge popultio hvig ukow sttistics µ, σ Emple of Sttisticl Ifeece Use to pedict µ. Questios: Is my smple dom smple? How close is my pedictio? How ceti is my pedictio? 7.3. Mesues of Cetl Tedecy Smple Me o Avege : N N i i Popultio Me o Avege µ : µ N N i i 89

90 Medi ~ : the middle vlue i k-odeed dt set Mode M : the dt vlue o sttistic tht occus most ofte Multi-Modl Dt Set: dt set with two o moe modes Medi Clcultio Pocess: Step : Rk ode fom smllest to lgest ll elemets i the dt set. Step : The medi ~ is the ctul middle sttistic if thee is odd ume of dt poits. Step 3: The medi ~ is the vege of the two middle sttistics if thee is eve ume of dt poits Mesues of Dispesio Rge R : R L S whee L is the lgest dt vlue i the dt set d S is the smllest dt vlue Smple Stdd Devitio s : N i. N i s Popultio Stdd Devitioσ : σ N i µ i N Smple Vice: s Popultio Vice: σ Smple Coefficiet of Vitio C VS : C VS Popultio Coefficiet of VitioC VP : Z-Scoe z i fo Smple Vlue i : z i C VP i s σ µ s 9

91 7.5. Smplig Distiutio of the Me The me is fomed fom smple of idividul dt poits domly selected fom eithe ifiite o fiite popultio. The ume of dt poits selected is give y. The smple is cosideed Lge Smple if 3 ; Smll Smple if < Epected Vlue of : E µ Stdd Devitio of : Ifiite Popultio σ σ Fiite Popultio of Cout N N σ σ N Lge Smple Z-scoe fo i : Whe σ is ukow, sustitute s. z i i µ σ / Itevl Estimte of Popultio Me: Lge-Smple Cse σ ± z α Smll-Smple Cse s ± tα Note: No ssumptio out the udelyig popultio eeds to e mde i the lge-smple cse. I the smll-smple cse, the udelyig popultio is ssumed to e oml o ely so. Whe σ is ukow i the lge-smple cse, sustitute s. σ Smplig Eo E R : E R zα Smple Size Needed fo Give Eo: z α σ. E R 9

92 7.6. Smplig Distiutio of the Popotio The popotio p is qutity fomed fom smple of idividul dt poits domly selected fom eithe ifiite o fiite popultio. The popotio c e thought of s me fomulted fom smple whee ll the idividul vlues e eithe zeo o oe. The ume of dt poits selected is give y. The smple is cosideed Lge Smple if oth p 5 d p Epected Vlue E X of p : E X p µ Stdd Devitio of p : Ifiite Popultio p p σ p Fiite Popultio of Cout N N p p σ p N Itevl Estimte of Popultio Popotio: p ± z α p p Note: Use p. 5 i p p if clueless o the iitil size of p Smplig Eo: p p E R z α Smple Size Needed fo Give Eo: z p p α ER Wose cse fo , popotio ukow: z α 4ER 9

93 Sectio II Tles 93

94 . Numeicl.. Fctos of Iteges though 9 The stdd ode-of-opetios pplies; ^ is used to deote the isig to powe; d * is used fo multiplictio. INTEGER FOLLOWED BY FACTORIZATION 9 Pime 57 3*9 Pime 3 *3*5 58 *9 3 Pime 3 Pime 59 Pime 4 ^ 3 ^5 6 ^*3*5 5 Pime 33 3* 6 Pime 6 *3 34 *7 6 *3 7 Pime 35 5*7 63 3*3*7 8 ^3 36 ^*3^ 64 ^6 9 3*3 37 Pime 65 5*3 *5 38 *9 66 *3* Pime 39 3*3 67 Pime ^*3 4 ^3*5 68 ^*7 3 Pime 4 Pime 69 3*3 4 *7 4 *3*7 7 *5*7 5 3*5 43 Pime 7 Pime 6 ^4 44 ^* 7 ^3*3^ 7 Pime 45 3^3*5 73 Pime 8 *3*3 46 *3 74 *37 9 Pime 47 Pime 75 3*5^ ^*5 48 ^4*3 76 ^*9 3*7 49 7*7 77 7* * 5 *5^ 78 *3*3 3 Pime 5 3*7 79 Pime 4 ^3*3 5 ^*3 8 ^4*5 5 5^5 53 Pime 8 3^4 6 *3 54 *3^3 8 *4 7 3^3 55 5* 83 Pime 8 ^*7 56 ^3*7 84 ^*3*7 94

95 Itege Followed By Fctoiztio 85 5*7 ^ 57 3*7^ 86 *43 *6 58 * *9 3 3*4 59 3*53 88 ^3* 4 ^*3 6 ^5*5 89 Pime 5 5^3 6 7*3 9 *3^*5 6 *3^*7 6 *3^4 9 7*3 7 Pime 63 Pime 9 ^*3 8 ^7 64 ^*4 93 3*3 9 3* *5* 94 *47 3 *5*3 66 * *9 3 Pime 67 Pime 96 ^5*3 3 *6 68 ^3*3*7 97 Pime 33 7*9 69 Pime 98 *7^ 34 *67 7 *5*7 99 3^* 35 3^3*5 7 3^*9 ^*5^ 36 ^3*7 7 ^*43 Pime 37 Pime 73 Pime *3*7 38 *3*3 74 *87 3 Pime 39 Pime 75 5^*7 4 ^3*3 4 ^*5*7 76 ^4* 5 3*5*7 4 3* *59 6 *53 4 *7 78 *89 7 Pime 43 *3 79 Pime 8 ^*3^3 44 ^4*3^ 8 ^*3^*5 9 Pime 45 5*9 8 Pime *5* 46 *73 8 *9 3* *7^ 83 3*6 ^4*7 48 ^*37 84 ^3*3 3 Pime 49 Pime 85 5*37 4 *3*9 5 *3*5^ 86 *93 5 5*3 5 Pime 87 *7 6 ^*9 5 ^3*9 88 ^*47 7 3*3*3 53 Pime 89 3^3*7 8 *59 54 *7* 9 *5*9 9 7*7 55 5*3 9 Pime ^3*3*5 56 ^*3*3 9 ^7*3 95

96 .. Pime Numes less th Rom Numel d Aic Equivlets ARABIC ROMAN ARABIC ROMAN ARABIC ROMAN I X CI II XI CC 3 III 5 XV 5 D 4 IV XX 6 DI 5 V 3 XXX M 6 VI 4 XL 5 V 7 VII 5 L L 8 VIII 6 LX C 9 IX C M 96

97 .4. Nie Elemety Memoy Numes NUM MEM NUM MEM NUM MEM φ π 3.46 l e.78 Loge Ameic Nmes fo Lge Numes NUM NAME NUM NAME NUM NAME ^3 thousd ^8 quitillio ^33 decillio ^6 millio ^ setillio ^36 udecillio ^9 illio ^4 septillio ^39 duodecillio ^ tillio ^7 octillio ^48 quidecillio ^5 qudillio ^3 otillio ^63 vigitillio.6. Selected Mgic Sques.6.. 3X3 Mgic Sque with Mgic Sum 5. The secod sque elow is clled 33 Ati-Mgic Sque:

98 .6.. 4X4 Pefect Mgic Sque with Mgic Sum 34: X5 Pefect Mgic Sque with Mgic Sum 65: Note: Fo Mgic Sque of size NXN, the Mgic Sum is give y the fomul N N 98

99 .6.4. Nested 5X5 Mgic Sque with Oute Mgic Sum 65: X6 Mgic Sque with Mgic Sum :

100 X7 Mgic Sque: Mgic Sum is Quduple-Nested 9X9 Mgic Sque with Oute Mgic Sum 369:

101 .7. Thitee-y-Thitee Multiplictio Tle Diffeet fot sizes e used fo, oe, two, o thee-digit eties Note: The shded locks o the mi digol e the fist thitee sques

102 .8. The Rdom Digits of PI The digits of PI pss evey domess test. Hece, the fist 9 digits of PI seve eqully well s dom ume tle. PI3.-- READ LEFT TO RIGHT, TOP TO BOTTOM

103 .9. Stdd Noml Distiutio THE STANDARD NORMAL DISTRIBUTION: TABLE VALUES ARE THE RIGHT TAIL AREA FOR A GIVEN Z Z Right Til Ae stts to fll elow. 3

104 .. Two-Sided Studet s t Sttistic TABLE VALUES ARE T SCORES NEEDED TO GUARANTEE THE PERCENT CONFIDENCE Degees of feedom: DF

105 .. Dte d Dy of Ye DATE DAY DATE DAY DATE DAY J My Sep 44 J 5 5 My 5 5 Sep 5 48 J 8 8 My 8 8 Sep 8 5 J My 3 Sep 55 J 5 5 My 5 35 Sep 5 58 J 9 9 My 9 39 Sep 9 6 J My 4 Sep 65 J 6 6 My 6 46 Sep 6 69 Fe 3 Ju 5 Oct 74 Fe 5 36 Ju 5 56 Oct 6 78 Fe 8 39 Ju 8 59 Oct 8 8 Fe 43 Ju 63 Oct 85 Fe 5 46 Ju 5 66 Oct 5 88 Fe 9 5 Ju 9 7 Oct 9 9 Fe 53 Ju 73 Oct 95 Fe 6 57 Ju 6 77 Oct 6 99 M 6** Jul 8 Nov 35 M 5 64 Jul 5 86 Nov 5 39 M 8 67 Jul 8 89 Nov 8 3 M 7 Jul 93 Nov 36 M 5 74 Jul 5 96 Nov 5 39 M 9 78 Jul 9 Nov 9 33 M 8 Jul 3 Nov 36 M 6 85 Jul 6 7 Nov 6 33 Ap 9 Aug 3 Dec 335 Ap 5 96 Aug 5 8 Dec Ap 8 98 Aug 8 Dec 8 34 Ap Aug 4 Dec 346 Ap 5 5 Aug 5 7 Dec Ap 9 9 Aug 9 33 Dec Ap Aug 34 Dec 356 Ap 6 6 Aug 6 38 Dec 6 36 ** Add oe dy sttig hee if lep ye 5

106 . Physicl Scieces.. Covesio Fctos i Allied Helth... Volume Covesio Tle Apothecy Household Metic miim dop gtt 6miims ml cc 6miims fluidm 6gtts tsp 5mL cc o 4mL 4fluidms.5fluidouce 3tsp tsp 5mL cc 8fluidms fluidouce tsp 3mL cc 8fluidouces cup 4mL cc 6fluidouces cups pit 5mL cc o 48mL 3fluidouces pits qut ml cc o 96mL... Weight Covesio Tle Apothecy Metic..3. Geel Commets gi 6mg o 64mg 5gis g 6gis dm 4g 8dms ouce 3g ouces poud 384g All thee systems pothecy, household d metic systems hve ough volume equivlets. Sice the household system is volume-oly system, the Weight Covesio Tle i.. does ot iclude household equivlets. Commo discepcies tht e still cosideed coect e show i itlics i oth tles.. d... 6

107 .. Medicl Aevitios i Allied Helth ABBREVIATION MEANING.i.d. Twice dy.i.w. Twice week c With cp, cps Cpsule dil. Dilute DS Doule stegth gtt Dop h, h Hou h.s. Hou of sleep, t edtime I.M. Itmuscul I.V. Itveous.p.o., NPO Nothig y mouth NS, N/S Noml slie o.d. Oce dy, evey dy p.o By o though mouth p... As eeded, s ecessy q. Evey, ech q..m. Evey moig q.d. Evey dy q.h. Evey hou qh Evey two hous q4h Evey fou hous q.i.d. Fou times dy ss Oe hlf s.c., S.C., s.q. Sucuteous stt, STAT Immeditely, t oce susp Suspesio t Tlet t.i.d. Thee times dy P stegth P gms pe ml A:B stegth A gms pe B ml 7

108 .3. Wid Chill Tle Gey e is the dge zoe whee eposed hum flesh will egi to feeze withi oe miute. T E M P F WIND SPEED mph Het Ide Tle The ume i the ody of the tle is the equivlet hetig tempetue t humidity T E M P F RELATIVE HUMIDITY

109 .5. Tempetue Covesio Fomuls.5.. Fheheit to Celsius: C F Celsius to Fheheit: F.8C 3.6. Uit Covesio Tle Aged i lpheticl ode TO CONVERT TO MULTIPLY BY ces ft 4356 ces m ces ods 6 ces hectes.447 ce feet els 7758 ce feet m Agstom å cm E-8 Agstom m. stoomicl uit AU cm.496e3 stoomicl uit km.496e8 tmosphees tm feet HO tmosphees i of Hg 9.9 tmosphees mm of Hg 76 tmosphees psi 4.7 tm.9869 dye/cm E6 psi l/i mm Hg 75.6 MP E- els l ft els m els gl US 4 els lite

110 TO CONVERT TO MULTIPLY BY BTU Cdi BTU.48 BTU cl BTU eg E- BTU joule cloie cl joule 4.84 cetimete cm ich.3937 cm m E- dcy m E-3 dye g cm /s dye Newto E-5 eg cl.396e-8 eg dye cm eg joule E-7 fthom ft 6 feet ft i feet m.348 fulog yd gllo US gl i 3 3 gllo lite Impeil gl i gllo lite gmm Guss E-5 gmm Tesl E-9 guss Tesl E-4 gm g poud.46 gm kg E-3 hecte ce.475 hecte cm E-8 hosepowe Wtt W 745.7

111 TO CONVERT TO MULTIPLY BY ich i cm.54 ich i mm 5.4 joule J eg E7 joule cl.396 kilogm kg g E3 kilogm poud.46 kilomete km m E3 kilomete ft kilomete mile.637 Kilomete/h kph mile/h mph.637 kilowtt hp.34 kot mph.5779 lite cm 3 E3 lite gl US.647 lite i mete gstom E mete ft mico cm E-4 mile ft 58 mile km.6934 mm Hg dye/cm 333. Newto dye E5 Newto poud foce.489 Newto-mete toque foot-poud-foce ouce l.65 Pscl tmosphees E-6 Pscl psi.45 E-4 Pscl to 7.5 E-3 pit gllo.5 poise g /cm/s poise kg /m/s.

112 TO CONVERT TO MULTIPLY BY poud mss kg poud foce Newto od feet 6.5 qut gllo.5 stoke cm /s slug kg Tesl Guss E4 To milli.3334 To millimete hg to log l 4 to metic l 5 to metic kg to shot o et l to shot o et kg to shot o et to metic.97 wtt J /s yd i 36 yd m.944 ye cled dys ye cled s E7.7. Popeties of Eth d Moo PROPERTY VALUE PROPERTY VALUE Distce fom 9..9^6 Eth 3. ft/s su miles Sufce g Equtoil Moo distce 38, miles dimete fom eth miles Legth of dy 4 hous Moo dimete 6 miles Moo 7 dys, 7 Legth of ye dys evolutio hous

113 .8. Metic System.8.. Bsic d Deived Uits QUANTITY NAME SYMBOL UNITS Legth mete m sic uit Time secod s sic uit Mss kilogm kg sic uit Tempetue Kelvi K sic uit Electicl Cuet mpee A sic uit Foce Newto N kg m s - Volume Lite L m 3 Eegy joule J kg m s - Powe wtt W kg m s -3 Fequecy hetz Hz s - Chge coulom C A s Cpcitce fd F C s kg - m - Mgetic Iductio Tesl T kg A - s Metic Pefies PREFIX FACTOR SYMBOL METER EXAMPLE pet ^5 E Em te ^ P Pm gig ^9 G Gm meg ^6 M Mm kilo ^3 k km hecto ^ h hm dec ^ d dm deci ^- d dm ceti ^- c cm milli ^-3 m mm mico ^-6 µ µ m o ^-9 m pic ^- p pm 3

114 .9. Bitish System.9.. Bsic d Deived Uits QUANTITY NAME SYMBOL UNITS Legth foot ft sic uit Time secod s sic uit Mss slug sic uit Tempetue Fheheit F sic uit Electicl Cuet mpee A sic uit Foce poud l deived uit Volume gllo gl deived uit Wok foot-poud ft-l deived uit Powe hosepowe hp deived uit Chge coulom C deived uit Cpcitce fd F deived uit Het Bitish theml uit Btu sic uit.9.. Ucommo Bitish Mesues of Weight d Legth WEIGHT LINEAR GiBsic Uit IchBsic Uit scuple gis hd4 iches dm3 scuples lik7.9 iches ouce6 dms sp9 iches poud6 ouces foot iches hudedweight pouds yd3 feet to pouds fthom yds log to4 pouds od5.5 yds chi liks yds fulog yds mile76 yds kot mile feet legue3 miles 4

115 .9.3. Ucommo Bitish Mesues of Liquid d Dy Volume LIQUID DRY GillBsic Uit PitBsic Uit pit4 gills qut pits qut pits gllo4 quts gllo4 quts peck gllos hogshed63 gllos ushel4 pecks pipe o utt hogsheds tu pipes.9.4. Miscelleous Bitish Mesues AREA sque chi6 sque ods ce43,56 sque feet ce6 sque ods sque mile 64 sque ces sque mile sectio towship 36 sectios ASTRONOMY stoomicl uit AU 93,, miles light secod 86, miles. AU light ye 5.88^ miles 6.36^4 AU psec pc 3.6 light yes kpcpc mpc pc VOLUME U.S. liquid gllo 3 cuic iches I Impeil gllo. U.S. gllos.6 cuic feet cod8 cuic feet 5

116 This Pge is Blk 6

117 Sectio III Applictios i Pesol Fice 7

118 . The Alge of Iteest.. Wht is Iteest? Iteest ffects just out evey dult i Ameic. If you e idepedet, ow c o home o oth, o hve cedit cd o two, you poly py o hve pid iteest. So, wht ectly is iteest? Iteest is et chge fo the use of moey. As et chge fo the use of housig ccumultes ove time, likewise, iteest chge fo the use of moey lso ccumultes ove time. Just s people sometimes oow housig whe shelte is eeded, people sometimes oow moey whe we wt o eed the items tht moey c uy. Iteest is omlly stted i tems of pecetge miles iteest te such s8 ye. Just s velocity 6 hou is te of distce ccumultio, pecetge iteest te is velocity of pecet ccumultio. Whe divig i Ameic, the customy uits of velocity e miles pe hou. Likewise, the customy uits fo iteest te e pecet pe ye. The ede should e we tht othe th customy uits my e used i ceti situtios. Fo emple, i spce tvel 7 miles sec is used to descie escpe velocity fom plet eth; d, whe computig cedit-cd chge, mothly iteest te of.5 moth my e used. Both velocity d pecetge iteest te eed to e multiplied y time specified i mtchig uits i ode to oti the totl mout ccumulted, eithe miles o pecet, s illustted elow. O the od: D miles 75 hou hous 75 3 miles I the k: moth pecet 3 moths 7 pecet $ Oce the totl ccumulted iteest is computed, it is the multiplied y the mout oowed, clled the picipl P, i ode to oti the totl ccumulted iteest chge I. 8

119 The totl ccumulted iteest chge I, the picipl P, the pecetge iteest te heefte, to e simply clled the iteest te, d the ccumulted time t clled the tem duig which fied piciple is oowed e elted y the Fudmetl Iteest Chge Fomul I Pt lso clled the Simple Iteest Fomul. This fomul pplies s log s the picipl P d the iteest te emi costt thoughout the time t. E..: Suppose $,. is oowed t 7 ye ove 4 moth peiod with o chge i eithe picipl o iteest te. How much e the totl iteest chges? Usig I P t, we oti fte covetig pecet to its fctiol equivlet d moths to thei yely equivlet 8 I $,. yes 3 yes $8.. I P t esemles the fomul D Rt, whee D is distce, R is costt velocity, d t is the Note: Notice how much the fomul time duig which the costt velocity is i effect. The vile P i I P t distiguishes the Fudmetl Iteest Chge Fomul i tht totl iteest chges e popotiol to oth the picipl oowed d the time duig which the picipl is oowed. Thee e two types of iteest: odiy iteest d ke s iteest. Odiy iteest is computed o the sis of 365 -dy ye, while kes iteest is computed o the sis of 36 -dy ye. The distictio usully shows up i shot dutio los of less th oe ye whee the tem is specified i dys. Give two ideticl iteest tes, picipls, d tems, the lo whee iteest is computed o the sis of kes iteest will lwys cost moe. E..: Suppose $ 5,. is oowed t 9 fo 5 ye dys. How much e the totl iteest chges usig A odiy iteest s the sis fo computtio, B kes iteest s the sis fo computtio? 9

120 Agi, usig I P t s ou fudmetl sttig poit, we oti 5 A I $ 5,.9 ye 365 yes $ B I 5,.9 ye yes $ $ 36 Notice kes iteest ets $ 64. to the k... Simple Iteest Simple iteest is iteest chged ccodig to the fomul I P t. We omlly fid simple iteest eig used i los whee the tem is eltively shot o the picipl is few thousd dolls o less. At oe time, simple iteest ws the iteest method pimily used to compute chges i utomoile lo. Tody, howeve, with some utomoile pices ppochig those of smll house e.g. the Humme my utomoile los e set up just like shote-tem home motgges. Whe we oow moey vi simple iteest cotct, ot oly e we to py the iteest chges, ut we lso must py ck the picipl oowed i full. Tht is the meig of the wod oowed: we e to etu the item used i the sme coditio tht it ws oigilly loed to us. Whe we oow moey, we e to etu it i its oigil coditio i.e. ll of it d with the sme puchsig powe. Sice moey ivily loses some of its puchsig powe with the pssge of time due to the effects of ifltio, oe c lmost lwys e sue tht the mout oowed is woth less t the ed of specified tem th t the egiig. Thus, y iteest chge levied must, s miimum, mke up fo the loss of puchsig powe. I ctulity, puchsig powe is ot oly peseved ut ctully icesed vi the pplictio of commecil iteest chges. Rememe, k is usiess d should epect pofit iteest o the sle of its pticul usiess commodity moey.

121 Retiig simple-iteest lo equies the pymet of oth the picipl oowed d the simple iteest chge icued duig its tem. Thus we c esily wite lgeic fomul fo the totl mout A to e etued, clled the Simple Iteest Fomul, A P I P P t P t. We c esily use the simple iteest fomul to help clculte the mothly pymet M fo y lo issued o the sis of simple iteest. E..: You oow $ 38,. fo SUV t 3.5 ye simple iteest ove tem of 7 yes. Wht is you mothly pymet? Wht is the totl iteest chge? : A P I P t $38,..35{7} A $47,3. : M 3 # A moths $47,3. $ : I A P $47,3. $38,. $9,3. Buyes should e we tht sometimes the ctul iteest te is moe th it is stted to e. A Simple Discout Note is type of lo whee this is ideed the cse. Hee, the oowe pepys ll the iteest up fot fom the picipl equested. Thus, the fuds F ville fo use duig the tem of the lo e i fct less, s give y the epessio F P I. This leds to hidde icese i iteest te if oe cosides the picipl to e those fuds F ctully tsfeed to the oowe. This et emple illusttes this commo sleigh-of-hd sceio. E..: A Simple Discout Note fo $,. is issued fo tem of 5 moths t. Fid the hidde iteest te. ye : I P t $,. ye yes $,5. : F P I $,. $,5. $87,5. 5

122 3 : I Ft,5. 87,5. 9,375.,5.,5. 9, ye 5 Notice tht the iteest te is icesed y. 4 pecetge poits y simply chgig the type of lo, i.e. Simple Discout Note. This will lwys e the cse: ot oly does iteest te mtte, ut lso the type of lo employig the iteest te. As show i ou lst emple, pecise fomuls llow oe to esily clculte the vious ficil qutities without esotig to the use of etesive ficil tles..3. Compoud Iteest The simple iteest fomul A P t is used i situtios whee the picipl eve chges duig the tem of the lo. But moe ofte th ot, the picipl will chge due to the fct tht ccued iteest is dded to the oigil picipl t egul itevls, whee ech itevl is clled compoudig peiod. This dditio cetes ew d elged picipl fom which futue iteest is clculted. Iteest duig y oe compoudig peiod is computed usig the simple iteest fomul. To see how this woks, let P e the iitil picipl d c e the iteest te duig the compoudig peiod e.g. fo ul iteest pplied vi mothly compoudig peiods, c. The fte oe compoudig peiod, we hve y the simple iteest fomul A P I P Pc P c P. Afte the secod compoudig peiod, we hve

123 A A P I P P c P c. P P P c c c Afte the thid compoudig peiod, the pocess cycles gi with the esult A A 3 3 P I P P P c c c P c P c 3. P 3 Lettig the pocess cotiue to the ed of compoudig peiods leds to the Compoud Iteest Fomul fo Totl Amout Retued A A P c. If is the ul iteest te d is the ume of compoudig peiods i oe ye, the the mout A fte tem of t yes is give y the fmili t compoud-iteest fomul A P. I ode to use eithe vesio of the compoud iteest fomul, o dditio to the iitil picipl P must occu othe th tht geeted y the compoudig effect duig the totlity of the compoudig pocess tem. The mout A is the mout to e etued whe the compoudig pocess is complete i.e. hs cycled itself though specified ume of compoudig peiods. Both fomuls e most commoly used i the cse whee iitil sum of moey is deposited i ficil/ivestmet istitutio d llowed to gow thoughout peiod of yes ude specified set of compoudig coditios. E.3.: A lump sum of $,. is deposited t 3 ye fo yes compouded qutely fou times pe ye. Fid the mout A t the ed of the tem. : A P A $,. A $,..75 t $34,

124 E.3.: A mout of $ 5,. compouds t peiod fo 4 peiods. Fid the mout A t the ed of the tem. : A P A $5,.. A $5,.. c 4 4 $7,33.84 E.3.3: A gdfthe ivests $5. i log-tem gowth fud fo his ewly-o gddughte. The fud is leglly iccessile util the child eches the ge of 65. Assumig effective iteest te of 9 ye compouded ully, how much will the gddughte hve ccumulted y ge 65? : A P A $5,. A $5,..9 t $,354,9.8 The lst emple shows the mgic of compoudig s it opetes o iitil picipl though log peiod of time. A eltively smll ficil gi eceived whe youg c gow ito mgificet sum if left to ccumulte ove sevel decdes. This simple ut poweful fct leds to ou fist Wods of Wisdom: If popely mged, youg widflls ecome old fotues..4. Cotiuous Iteest t Coside the compoud iteest fomul A P. Wht would e the ovell effect of icesig the ume of compoudig peiods i oe ye while holdig oth the ul iteest te d the tem t costt? Oe c immeditely see tht the epoet t would gow i size, ut the qutity iside the petheses,, would ecome lmost idistiguishle fom the ume s iceses idefiitely. 4

125 Sice o mtte how lge is, the dimiishig of to my egte the effect of hvig lge d lge epoet. Thus, we ed up with mthemticl tug of w etwee the two t ffected qutities i A. Ou epoet is gowig lge despetely tyig to mke A idefiitely lge ume. By cotst, ou se is eig the ume tyig to mke A. Which wis? O, is thee compomise? whee To eploe this issue, we ll fist look t specific emple 5, t yes, P $., d, susequetly, ye.5 A $.. The ume of compoudig peiods i ye will e llowed to icese though the sequece,,,, 365,,,,,, d,,. Mode clcultos llow clcultios such s these to e esily pefomed o outie sis. The esults e displyed i the tle elow with the coespodig mout geeted y usig the simple iteest fomul A P t. A $ $ $ $ $ $ $ $.6487 $.6487 Notice tht s pogessively iceses without oud, the mout A ecomes moe d moe ceti, stilizig out oe digit to the left of the deciml poit fo evey powe of te. I coclusio, we c sy tht the ttle eds i tidy compomise with < A <, i pticul A

126 The pocess of pogessively icesig without oud is clled limit pocess d is symolized y the limit symol lim. Limit pocesses e etesively used to deive most of the mthemticl tools d esults ssocited with clculus. We ow ivestigte A s fo the cse of fied ul iteest t te d tem t i yes, A lim[ P ]. To lyze this epessio, we fist move the limit pocess iside the petheses d et to the pt of the epessio it diectly ffects to oti t A P{ lim[ ]}. Agi, we hve set up ou clssic ttle of opposig foces: the epoet gows without oud d the se gets eve close to. Wht is the comied effect? To swe, fist defie m m. Fom this, we c estlish the towig eltioship m. Sustitutig, we oti t A lim[ P ] A P{lim[ A P{lim[ m m m ]} t t ]}. m Now ll we eed to do is evlute lim[ ], d we will do m this evlutio the mode, esy wy, vi scietific clculto. m vlue m m m 6

127 We will stop the evlutios t m,,. Notice tht ech time m is icesed y fcto of, oe moe digit i the m epessio m is stilized. If moe deciml plces e m eeded, we c simply compute m to the ccucy desied. m Whe m gets stoomiclly lge, the epessio m coveges to the ume e Coespodigly, ou fil limit ecomes A P A P{ e} A Pe { lim[ m m t t m ]} t. t The lst epessio A Pe is kow s the Cotiuous Iteest Fomul. Fo fied ul iteest te d iitil deposit P, the fomul gives the ccout lce A t the ed of t yes ude the coditio of cotiuously ddig to the cuet lce the iteest eed i twiklig of eye. The cotiuous iteest fomul epesets i itself uppe limit fo the gowth of ccout lce give fied ul iteest te. Hece, it is vey impott d esily used tool, which llows peso to quickly estimte ccout lces ove log peiod of time. The followig emple will illustte this. E.4.: A iitil deposit of $,. is compouded mothly typicl tuove fo compy 4K ccout, etc. t 8 fo ye peiod of 3 yes. Compe the fil mouts otied y usig oth cotiuous d compoud iteest fomuls. : A Pe A $,.e.8 3 A $,3.76 t 7

128 : A P A $,. A $,..667 A $9, t Notice tht thee is less th $ 4. diffeece etwee the two mouts, which shows the cotiuous iteest fomul vey vlule tool fo mkig estimtes whe the ume of compoudig peiods i ye eceeds twelve o moe. By povidig quick uppe oud fo the totl mout to e etued, the cotiuous iteest fomul c lso e thought of s fiscl gold stdd defiig the limitig cpilities of the compoudig pocess. I the et two emples, we eploe the use of the cotiuous iteest fomul i povidig pid estimtes fo oth iteest te d time eeded to chieve give mout A. I ech emple, the tul logithm deoted y l is fist used to elese the ovell epoet i e t, which, i tu llows oe to solve fo eithe o t. E.4.: A okege house clims tht $,. is guteed to ecome $,,. i 4 yes if left with them. Wht iteest te would mke this so? : A Pe $,e e 4 4 : l e 4 4 l e l t Pe.57.5 t A $,,. l ye 8

129 The iteest te of.5 ye my e otile, ut epesets ggessive estimte sice the vege Dow-Joes-Idustil- Avege ul te of etu hs hoveed oud 9 ye fo the lst 4 yes. Hece the ochue is mkig mkete s clim! Suppose we ctively mged ou ccout fo 4 yes whee we wee ctully le to chieve 9 ye. The.9 4 A $,.e $365,98.34, which is tidy sum, ut o millio. Let uyes ewe, o, ette yet, let uyes e le to figue fo themselves. E.4.3: How log does it tke sttig picipl P to quduple t 5 compouded mothly? : A Pe 4P Pe Pe.5 t : l e.5 t ye 4P e.5 t.5t.3869 t t 7.73yes.5 t l Effective Iteest Rte How do we compe oe iteest te to othe? The questio ises sice ot oly does ctul iteest te mtte, ut lso the wy the te iteest is utilized i.e. type of compoudig mechism. The effective ul iteest te, desigted eff, povides mthemticl sis fo compig iteest tes hvig diffeet compoudig mechisms. eff is defied s tht ully-compouded iteest te tht geetes the sme mout s the specified iteest te d ssocited compoudig pocess t the ed of t yes. I the cse of the compoud iteest fomul, we hve 9

130 P eff eff eff eff t t P[ [ ] t ] t. I the cse of cotiuous iteest, we hve P eff eff eff e eff e t t [ e Pe ] t t I the cse of simple iteest, we hve P eff eff eff t eff t t t P t t t t The effective iteest te, s defied ove, is simple d poweful cosume sis of compiso i tht it comies oth te d pocess ifomtio ito sigle ume. Bks d othe ledig istitutes e leglly equied to stte effective iteest te i thei dvetisig d o thei documets. Stock mket etus ove log peiod of time e omlly specified i tems of vege ul gowth o iteest te. We defiitely eed to kow the meig of eff d its use if we e to suvive the cofusio of umes tossed ou wy i mode society. 3

131 E.5.: Which is the ette del, 7.5 ye compouded cotiuously o 7.5 compouded qutely? : eff eff eff e : eff e.75 ye The ette del is effective iteest te is 4 ye ye R eff 7.73 ye ye 7.5 compouded qutely whee the. Whe viewed s geel cocept, the effective ul iteest te ecomes poweful ecoomic d foecstig tool i tht it c e esily dpted to detemie the vege ul gowth te fo secuities o y pheome whee chge occus ove peiod of yes. E.5.: Secuities vlued t $ 5,. i 98 hve gow i vlue to $ 8,. i 5. Assumig cotiutio of the vege ul gowth vlue s ledy displyed duig the pst 5 yes, poject the vlue of these sme secuities i 45. Digmmig the polem i two steps, we hve 5 yes : P $5, yes : P $8,. 545 A $8,. A? Utilizig the geel defiitio of eff s foud i A P eff llows us to esily solve this polem fo ech step. t 3

132 : A P $8,. $5,. eff 5 eff eff : A $8,..7 A $8,..7 t eff ye eff $,3,4. The vege ul iteest/gowth te of.7 ye is vey good d shows ctive mgemet of the ovell gowth pocess. The fil ewd, $,3,4., is well woth it! E.5.3: A pofessiol s sly gows fom $9949. to $ 7,95. ove peiod of 3 yes. Wht is the vege ul gowth te? 3 yes Digmmig: P $9,949. A $7,95.. Solvig, we hve : A P $7,95. $ eff eff 3 eff eff t ye The fil vege gowth te of 8.7 ye cetily eceeds the vege ifltio ul te of 3 ye d shows stedy icese i puchsig powe ove time. eff 3 3

133 E.5.4: $,. is let to fied t ye simple iteest fo peiod of 5 yes. Wht is eff? : eff eff eff 5 t.9 t.5 ye Q 5..9 E.5.5: You hve $5, to ivest fo yes. Which of the followig thee dels is most dvtgeous to you, the ivesto: ye simple iteest fo the etie time peiod, 7 ye iteest compouded dily fo the etie time peiod, o 8 ye iteest compouded qutely fo the etie time peiod? We lyze polem i two stges. Fist, we will compute the eff fo the thee cses otig tht dily iteest 365 compoudig peiods ye is fo ll effects d puposes idistiguishle fom cotiuous iteest. The highest eff will the povide ou swe. Secodly, i the mode spiit of show me the moey, we will compute the epected eigs i ll thee cses. Compig eff : eff : 3 : eff eff eff. e ye 8.43 ye ye Qutely compoudig t 8 ye povides the est del. Clcultig the ssocited epected eigs gives 33

134 : A P t A $5,. [.] $55,. lt : A P A $5,..84 : A Pe A $5,.e lt : A P A $5, : A P A $5,. 3lt t eff eff.7 $5, : A $5,..843 t t t.8 4 $55,.3 $5, $55,.99 $55,99.89 Cse thee, qutely compoudig t 8 ye, hs the highest epected eigs s pedicted y the ssocited eff. The thee ltete clcultios use the effective ul iteest-te costuct fomul to ive t the ect sme swes withi doll o two s those poduced y the ssocited compoudig fomuls. This would e epected; fo this is how the thee eff fomuls wee deived i the fist plce! 34

135 . The Alge of the Nest Egg.. Peset d Futue Vlue Moey chges its vlue with time. This fct is s ceti s the poveil deth d tes. Ifltio is foce eyod idividul s cotol tht lesses the vlue of moey ove time. Smt ivestig coutes ifltio i tht it ehces the vlue of moey ove time. The vlue of moey ight ow is clled the peset vlue PV. The time-chged equivlet vlue i the futue is clled the futue vlue FV. This c e digmed s PV pocess time FV. I ode fo peset vlue to ecome futue vlue, oth time d pocess eed to e specified. This is ectly the cse i the t fmili compoud iteest fomul A P. Usig the ove geel digmmtic ptte, we c digm the compoud-iteest fomul s follows Replcig t P A. t P & A with PV & FV espectively leds to t PV FV. t Note: The ove fomul is ot completely coect util oe tkes i ccout the effects of ifltio, lysis optio. To ccout fo ifltio, sutct the ul ifltio te fom the give ul iteest te. Use the modified te i peset-to-futue vlue fomuls to poject ifltio-djusted futue vlue. 35

136 With this lst ote i mid, we peset the fou coupled Peset-to-Futue-Vlue Fomuls. All iteest tes i the fomuls elow eed to e ifltio djusted pe dj i if oe wts to oti ifltio-djusted futue vlue. Compoud Iteest: Effective Iteest: Cotiuous Iteest: Simple Iteest: FV PV FV PV eff t t FV PVe PV FV PV t t FV PV FV t e FV FV PV t PV t eff t Notice tht the coupled Peset-to-Futue-Vlue Fomuls llow us to esily move fom peset vlue to futue vlue o vis ves s log s the compoudig pocess, time peiod, d oe of the two vlues peset o futue is specified. Coupled peset-tofutue-vlue fomuls llow us to estimte totl chge i moety vlue s eithe ivestmets o dule goods move fowds o ckwds i time ude give set of pocess coditios. E..: Bill wishes to hve $,8,. i his Idividul Retiemet Accout IRA whe he eties i 35 yes. Wht is the peset vlue of this mout ssumig vege ul compoudig te of.5? : PV FV eff ye $,8,. PV 35.5 $,8,. PV $39, t 36

137 E..: Repet the clcultio i E... if vege ifltio te i 3 cts though the sme 35 ye time peiod. ye Bill s wish c e estted i tems of uyig powe. Wht Bill elly wts is $,8,. i cuet uyig powe y the time he eties i 35 yes. Thus : FV PV t FV PV i eff FV $5,64,95.4 FV $,8,..3 t 35 Itepeted, $ 5,64,95. 4 is the mout eeded 35 yes fom ow just to peseve the uyig powe iheet i $,8,. tody ssumig log-tem stedy ifltio te of i 3. Tuig to the peset vlue of this ew mout ye ssumig the sme.5, we hve : PV FV eff ye $5,64,95.4 PV 35.5 $5,64,95.4 PV $96, t Whe ifltioy pice iceses fo dule goods e stted i tems of ully-compouded pecetge jump, we typiclly use peset-to-futue-vlue fomuls to estimte the futue pice. This is especilly tue fo sigle ig ticket items such s houses, cs, ots, jewely, etc. Ou et emple illusttes the use of peset-to-futue vlue fomul to estimte the futue pice of ewly-uilt house. 37

138 E..3: The pice of ew house i ceti city iceses t vege te of 5. If pticul 3-edoom model i ceti ye sudivisio is piced t $ 35,. i 6, estimte the pice of simil model i the sme sudivisio i. : FV PV eff FV $85,644. FV $35,..5 t 4 This is some discocetig ews i tht the sme house will sell fo ppoimtely$ 85,644. fou yes fom ow. If you c ffod it, you ette uy ow. Witig costs moey! E..4: Clculte the peset vlue of $,. copote od comig due i 5 yes t 5 compouded qutely. ye FV : PV t $,. PV $47, If edeemed tody, the od would fetch $47, Gowth of Iitil lump Sum Deposit If iitil lump-sum deposit is the oly mes y which moety gowth is chieved, the the Peset-to-Futue-Vlue Fomuls e sufficiet to pefom the ssocited clcultios. We eed oly to idetify the pocess y which the gowth is occuig: ul compoudig vi effective iteest te, cotiuous compoudig, o compoudig fo fiite ume of compoudig peiods pe ye. Ech compoudig pocess hs ssocited fomul to which totl time d iteest te must e supplied. 38

139 E..: Wht is the futue vlue o-ifltio djusted t ge 65 of $ 3,. ivested t ge 5 ssumig eff 8 ye thoughout the 4-ye tem? Note: the mkig of moety-gowth digm is stogly ecommeded s fist step fo ll peset-to-futue-vlue polems sice pictues egge the use of oe s ight i d the ssocited sptil polemsolvig cpilities. Hece, fo Emple.., the ssocited moetygowth digm is eff 8 ye :$3,. FV? ge5 ge65 Solvig: : FV PV eff FV $8,47.77 FV $3,..8 t 4 E..: Clculte the effective ul iteest te eeded to tu $,. ito $,,. ove 5 ye peiod. t eff? :$,. $,,. t 5 Note tht the pocess mechism implicitly ssumed is ul compoudig vi the efeecig of ukow eff. Solvig: :$,,. $,. 5 eff eff eff. eff 5.. ye eff 5 39

140 The effective ul iteest te of eff. ye is poly impossile to susti fo eteded peiod of 5 yes. Eve i the go-go high-tech 9s, tes of this mgitude lsted fo oly si yes o so. E..3: Wht cotiuous iteest te is eeded to quduple give peset vlue i 5 yes? Askig fo cotiuous iteest te cot mes tht the cotiuous iteest fom of the peset-to-futue vlue fomul t FV PVe should e used. Also, the polem sttes tht the equied futue vlue is FV 4PV. Aottig this ifomtio o the moety-gowth digm d solvig gives t cot cot? : PV 4PV t 5 5 :4PV PVe 5 4 e 5 l4 l e ye The stted cotiuous iteest te of 9.4 ye is cetily chievle i tody s mkets; howeve, it is ot utomtic d will equie ctive mgemet of oe s ivestmets. Ou lst emple illusttes wht hppes if moe th oe deposit is mde duig the ovell ivestmet peiod. E..4: Wht is the pojected futue vlue igoig ifltio of etiemet fud whee iitil deposit of $ 4,. is mde t ge 3 d susequet deposit of $ 6,. is mde t ge 4. Assume effective ul iteest te of d ticipted etiemet ge of 68. eff ye 4

141 Udestdly, the moety-gowth digm iceses i compleity s it is modified to show the $ 6,. deposit o isetio ito the ivestmet pocess t ge 4. Agi, y the sttig of effective ul iteest te eff, the moetygowth pocess is udestood to e ul compoudig. :$4, ge3 eff ye FV? ge68 $ ge 6, 4 Solvig fo the pojected futue vlue equies diect dditio of two lgeic tems. : FV $4, FV $4,. 38 FV $,36, $6,. $6, FV $,496,73.73 $865,59.6 eff To summize E..4, $,. ivested y the ge of 4 ecomes $,36, y ge 68 if the stted coditios hold thoughout the ivestmet peiod. Suppose tht i E 4..4 sigle deposit could e mde t ge 3 i ode to cete the sme $,36, y ge 68. How much would such deposit e? By diect pplictio of the coupled Peset-to-Futue Vlue Fomuls $,36, PV $63,3.59, 38. et svigs to the ivesto of $36, Clcultig the ifltio-djusted futue vlue of $,36, ove the sme 38 yes, we oti $,36, FV dj $767, eff 8 4

142 .3. Gowth of Deposit Stem Most of us do t hve iitil lump sum of $ 4,. o$ 63,3. 59 y which to uild etiemet fud. The moe typicl wy we uild ou etiemet fuds is y mes of peiodic deposit eithe though pyoll deductio o diect self-disciplie tht ccumultes i vlue ye fte ye. Ad, fte thity yes o so, we e tlkig out sum jokigly efeed to s el moey. But it is o joke o how the sum is otied: though disciplie, scifice, d ttetive moey mgemet. I this sectio, we will develop d use the equtios tht detemie the futue vlue of egul deposit stem ove eteded peiod of time. Let D i D : i, t e deposit stem of ideticllysized pymets mde ove peiod of t yes whee is the ume of compoudig peiods pe ye d is the ul iteest te. Suppose tht ech deposit D i is sequeced to coicide with the egiig of the coespodig compoudig peiod d tht the lst deposit D t egis the lst of the t compoudig peiods. Ude these coditios, wht is the futue vlue of the etie deposit stem? Digmmig, D FV?. D Now, ech deposit D3 D4 D5 D t D i cotiutes potio t i vlue FV whee D t FV i D i. Thus, FV i to the totl futue FV FV FV t FVi i t t i Di. i t t i D i 4

143 The epessio t t { } t i D... D i is geometic seies d c e summed ccodigly s { } D D t i ti t D t leds to the followig fomul: FV { } Suppose we wt to coclude ou tem of t yes with oe fil deposit D t s show i the modified deposit stem D FV? D D3 D4 D5 D t D t D t To do so, dd oe moe D to oti I the cse of ul compoudig whee yely totl o te, the two fomuls ecome, t { } D FV. D t Without Fil Deposit: FV D No Fil Deposit: FV eff eff d D is eff { } eff t { } Simil fomuls e developed fo the cse of cotiuous compoudig i Sectio III, Topic 5. As discussed peviously, ll futue vlues must e djusted fo ifltio i ode to sceti tue uyig powe. eff eff. 43

144 E.3.: Afte tem of 3 yes, wht is the pojected futue vlue of etiemet fud whee 3 ul deposits of $ 5. e fithfully mde o Juy of ech succeedig ech ye. Assume. eff ye A modified moety-gowth digm c e used to show the peiodic ul deposits s follows: t eff :$5,. 9 ye FV? $ 5. t Hee, the digm stts with the fist ul deposit of $ 5. t t d ottes vi multiplictio the susequet 9 ul $ 5. deposits mde t the stt of ech ul compoudig peiod. Solvig, D t FV { eff eff } eff $5. FV. FV $,4, {.. } To summize, 3 ul deposits of $ 5. totlig $ 5,. hve gow to futue vlue of $,4, ove 3-ye tem ssumig eff ye. E.3.: Suppose sigle lump-sum deposit could e mde t the stt of the 3-ye peiod i Emple.3. i mout sufficiet to cete the sme futue vlue of $,4, How much would e eeded? Assume. t eff ye : PV? $,4, t 3 eff. ye 44

145 : PV PV. 3 eff t FV $,4, $,4, PV $48, E.3.3: Sm cotiutes $. pe moth to college svigs ccout fo his dughte My, who just tued. I dditio, he mkes ous deposits of $. o My s ithdy. Sm stted this pctice with comied $. deposit o the dy of My s ith d will csh out o My s 8 th ithdy with fil deposit of$.. How much will e i My s college svigs ccout t tht time ssumig 7 ye d mothly compoudig? This polem c e thought of s two su-polems: mothly deposit stem of 7 idividul deposits ove tem of 8 yes d pllel yely deposit stem of 9 idividul deposits ove peiod of 8 yes. The totl futue vlue will e the sum of oth pllel deposit stems the dy My tus 8. Fo the mothly deposit stem, we slightly modify the moetygowth digm to show the iclusio of the fil deposit. t 7 ye :$. 5 Solvig: D : FVmoth $. FVmoth.7 FV $86,846.7 moth $. FV t 8 $. t { }.7 7 { } moth? 45

146 Fo the yely deposit stem, we will fist eed to compute the.7 effective ul iteest te: eff 9 7. ye. Now, we hve ll the ifomtio eeded to compute FV ye d, cosequetly, FV FV FV : FV totl :$. 7 FV FV FV ye ye totl t ye eff 7.9 ye $. $..79 $38,68.93 FV D eff moth moth t { } ye $. FV 9 {.79 } FV eff ye t 8 $5,5.54 Ech of the fou deposit-stem fomuls c lso e used to detemie the peiodic deposit D eeded i ode to ccumulte specified futue vlue ude give set of coditios. E.3.4: Suppose Sm is ot hppy with the $ 5,5. 64 ccumulted y My s 8 th ithdy d, isted, would like to ccumulte futue vlue of $ 6,. vi the sigle mechism of mothly deposits. A How much should this deposit e, gi, ssumig mothly compoudig d 7 ye? B Wht sigle lump-sum deposit mde o the dy My ws o would geete equivlet futue vlue? ye? A t 7 ye : D? 5 D? FV? D? t 8 46

147 : FV D D FV t { } t { }.7$6,. D D $ B.7 7 { } t 7 ye t 8 : PV? $6,. : PV t PV.5833 FV 6 $6,. PV $6,. $45,55.9 This emple suggests the old mim of py me ow o py me lte. Oe could thik of ow s sigle pymet of $ 45,55. 9 d lte s deposit stem of 7 pymets, ech$ , totlig $79, The Two Gowth Mechisms i Cocet Sometimes, we my hve the oppotuity to ope up etiemet o college ivestmet ccout with espectle lumpsum deposit deote y L S pehps gied y wiig lottey o eceivig iheitce. Fom the o, we cotiute to this deposit y mes of deposit stem s show i the moetygowth digm L FV?. S D D3 D4 D5 D t D t D t 47

148 If LS > Di D which would suely e the cse fo 99 of the time, the we could edw the moety-gowth digm s follows L S D D FV?. D D3 D4 D5 D t D t D t Emiig this lst digm, oe lgeic epessio c e esily witte fo the ssocited futue vlue y summig the two emedded moety-gowth pocesses: t { } t D FV LS D. E.4.: Suppose Bill mkes qutely deposits of $. to etiemet fud ove peiod of 35 yes tht is stted with iitil deposit of $ 5. d cocluded with fil deposit of $.. Wht is the futue vlue ssumig qutely compoudig d 8? ye The moety-gowth digm iceses i compleity gi. t 35 t $. FV? 8 :$3. $. 38 Solvig: : FV L D FV $3. S D ye $. t { } t { } $, FV $47, $,53, $,579,

149 Note: The ede my sk, Is this the oly wy tht moety-gowth digm c e dw? The swe is emphtic o! These digms e offeed s suggested ppoch fo two esos: they visully imply flow of moey d they hve ee clssoom tested. The impott thig is to mke moety-gowth digm tht hs meig to you d upo which you c ssemle ll the elevt ifomtio. E.4.: Suppose i E.4., Bill stts his etiemet ccout o his 5 th ithdy d stops cotiutig o his 6 th ithdy with pls ot to withdw fom his ccout util the ge of 7. Bill is ecomig icesigly wy of highe-isk ivestmets s he gows olde. Hece, Bill olls his etiemet ccout ove ito sfe U.S. govemet-od fud pyig effective ul iteest te of eff 4.5 o his 6 th ithdy. Wht will e the ye futue vlue of Bill s etiemet ccout t ge 7? :$,579,68.9 t : PV eff $,579, FV FV $,453,5.4 t eff ye FV? t FV E.4. illusttes the impotce of eig le to choose the ight fomul fo the ight sceio. I my ivestmet sceios, sevel fomuls my hve to e used i ode to oti the sought-fte swe. Udestdig of the udelyig cocepts d fcility with lge e the two keys to success. We will ow list ll fou futue-vlue fomuls with iitil lump sum deposit L S coespodig to the fou deposit-stem fomuls. Fil Deposit & Othe-th-Aul Compoudig: t D t FV LS D { } 49

150 No Fil Deposit & Othe-th-Aul Compoudig: t D t FV LS D Fil Deposit d Yely Compoudig: t D FV LS D eff { } t { } No Fil Deposit d Yely Compoudig: t D t FV LS D eff eff eff eff eff { } Ou et emple illusttes the itegtio of mid-life widfll ito oe s etiemet pogm. E.4.3: Geoge gdutes fom usig school t ge d ccepts sig-o ous of $ 7. to go to wok t locl hospitl. At the time, Geoge used $. of the moey to ope up Roth IRA see Sectio I: 6.9. He cotiutes $. pe ye mkig the fil deposit t ge 6. Geoge is fily stute ivesto d ws le to chieve effective ul iteest te of eff.5 ye ove the couse of 38 yes. Additiolly, t ge 45, Geoge eceived smll iheitce of $ 5,. tht he pomptly ivested i t-fee muicipls pyig effective ul iteest te of 4.5. Wht e Geoge s totl holdigs t ge 6? eff ye Fo the Roth potio :$. $. 38 ge6 ge $. FV Roth? eff.5 ye $. eff 5

151 : FV FV FV FV Roth Roth Roth Roth L S D { } t D t eff 39 { } 38 $. $ $. $ $87,86.94 $78, $87,69.4 eff eff 39 { } Fo the t-fee-muicipls potio :$5,. FV : FV FV FV tfee tfee ge45 tfee PV eff 4.5 ye t $9,9.3 eff tfee ge6 $5,..45 5? Filly: FV FV Roth FV $899, tfee To ecp, though smt ivestig, Geoge ws le to tu cotiutios totlig $ 55,. ito $ 899, ove 38- ye peiod..5. Summy This ticle is ot iteded to e tetise o etiemet plig. All seious etiemet plig should stt with licesed ficil cosultt i ode to devise detiled log-tem ctio pls tht meet idividul gols. The impott thig i this dy of ge is to just do it! This leds to secod Wods of Wisdom: You must fist pl smt! The, you must do smt i ode to chieve tht coveted ecoomic secuity! 5

152 We close this ticle with the tle elow, poweful motivtiol id tht shows the futue vlue of $ 4. yely deposit fo vious tems d effective ul iteest tes. Notice tht the shded millio-doll levels c e eched i fou of the five colums. Rechig et woth of oe millio dolls o moe is mtte of oth time d vege ul iteest te. The fomul used to costuct the tle is eff t { } D FV eff. GROWTH OF $4. YEARLY DEPOSIT EFFECTIVE ANNUAL INTEREST RATE TERM y $7,7 $8,63 $3,93 $3,65 $33,9 y $56,87 $63,34 $7,4 $78,45 $87,57 5 y $94,69 $,55 $3,3 $56,759 $86,686 y $4,877 $79,46 $7,58 $89,6 $369,879 5 y $4,453 $74,75 $373,95 $5,995 $77,4 3 y $83,43 $48,9 $598,3 $887,65 $,39,6 35 y $383,345 $595,653 $944,498 $,5,657 $,474,997 4 y $5,359 $858,438 $,477,67 $,587,37 $4,585, y $674,74 $,7,7 $,96,744 $4,384,675 $8,475,4 5 y $883,6 $,743,943 $3,557,764 $7,43,343 $5,64,97 5

153 . The Alge of Cosume Det 3. Lo Amotiztio Vey few people uy house with csh. Fo most of us, the motgge is the time-hooed wy to home oweship. A motgge is log-tem colltelized lo, usully with ficil istitutio, whee the title-deed to the house itself is the colltel. Oce motgge is secued, motgge pymets e the mde moth-y-moth d ye-y-ye util the mout oigilly oowed is fully pid, usully withi pe-specified time i yes. We cll this pocess of methodiclly pyig ck pymet y pymet the mout oigilly oowed motizig lo. The wod motize mes to liquidte, etiguish, o put to deth. So, to motize lo mes to put the lo to deth. I geetios pst especilly those i the Getest Geetio, the fil pymet i puttig lo to deth ws celeted with the ceemoil uig of some of the motgge ppewok. This symolized the deth of the motgge d the ssocited tsfeece of the title deed to the poud d det-fee homeowes. Nowdys, we By Boomes o Geetio Xes do t usully hg o to motgge log eough to hve the stisfctio of uig it. Suppose we oow motgge mout A, which is scheduled to e compouded mothly fo tem of T yes t ul iteest te. If o pymets e to e mde duig the tem, d sigle lloo pymet is to e mde t the ed of the tem, the the futue vlue FV of this sigle lloo pymet is A T FV A A. Now, let D Di : i, T e stem of ideticlly-sized motgge pymets mde ove the sme tem of T yes whee the fist pymet is mde ectly oe-moth fte motgge iceptio d the lst pymet coicides with the ed of the tem. 53

154 The, the totl futue vlue stem is D FV D FV D ssocited with the pymet T { } Fo the motgge to e pid, the futue vlue of the motggemout oowed must e equl to the totl futue vlue of the motgge-pymets mde. Hece, FV A D FV D T { } A A D D FV T D T { } A T { } FV A T. The lst epessio is the mothly pymet D eeded to motize motgge mout A t the ed of T yes give fied ul iteest te. Oce D is detemied, we c compute the peset doll vlue of the etie pymet stem PV PS TD d the peset doll vlue of ll the iteest pid vi the etie pymet stem TD A. PV IPS Aothe fudmetl qutity ssocited with lo, pticully motgge lo, udegoig the pocess of motiztio is the ctul doll vlue of the oigil lo still upid clled the pyoff o pyout vlue fte give ume j of mothly pymets D hve ee mde. We will deote this pyoff vlue y the lgeic symol PO. j 54

155 th th Recll tht the j pymet is mde t the ed of the j compoudig peiod. By tht time, the mout oowed will hve gow vi the compoudig mechism to A j. I like fshio, the futue vlue of the fist mothly pymet D will hve j gow to D, d the totl futue vlue of the fist j { } j D. Hece, the mout of the pyoff PO j tht coespods to ectly the fist j mothly pymets D is mothly pymets D will hve gow to j { } j D PO j A. Fo y fied motiztio tem T, the pyoff mout udegoes st th egtive chge fom the j pymet to the j pymet s it is icemetlly educed thoughout the life of the lo. The th egtive of this chge is the ctul doll mout D Aj of the j pymet ctully pplied to lo eductio o to picipl, see et ote. Thus, D D A D D D Aj Aj Aj Aj Aj A PO PO D j { } j { } j j A j j D j [ ] D D A D A j j PO [ ] j j PO j j [ ] j 55

156 Filly, the doll mout of the the pymet of iteest I is th j pymet D goig towds D D. Ij D Aj Note: I this hd ook, we hve delietely shied wy fom the tem picipl i fvo of moe use-fiedly tems tht llow the costuctio of o-ovelppig d peumoic lgeic symols. Tditiolly, the picipl P is cpitl sum iitilly oowed o iitilly deposited to which compoudig mechism is pplied. The si lo-motiztio fomuls peseted thus f c e split ito two goups: Glol Amotiztio Fomuls d Pymet Specific Fomuls. Oe must fist compute the mothly pymet D i ode clculte ll emiig qutities i eithe goup. Glol Amotiztio Fomuls Mothly Pymet: A T { } D Sum of Pymets i Pymet Stem: PV PS TD Totl Iteest Pid i Pymet Stem: PV IPS TD A Pyoff fte the PO Pymet Specific Fomuls th j Mothly Pymet: j A D j { } j Amout of Amout of th j Mothly Pymet to Picipl: D Aj D A j th j Mothly Pymet to Iteest: D Ij D D Aj 56

157 E 3..: A $ 4,. usiess-impovemet lo is egotited with locl k fo iteest te of 7 ye d motiztio tem of 7 yes. Fid the qutities D, PV PS, PV IPS, PO 8, D A, d D I. Sice these si qutities e diect sigle-step pplictio of the ssocited fomuls, pocess digm is ot eeded. A : D T { }.7 $4,. D.7 4 { } D $ mo : PV PV PV 3 PS PS : PV PV PV IPS IPS PS TD 7 $ $685,63.9 IPS TD A $685,63.9 $4,. $85,63.9 The lst thee qutities e pymet specific. 4 j D j : PO j A { } PO 8 $4,. $75, { } $ PO $,39, $,64,56.4 PO

158 PO 8 is lso the lloo pymet eeded i ode to motize the lo yes hed of schedule t the ed of 5 yes. 5 D A j : DAj $ $4, DA D $ A : DI D DA DI $ $834.4 D $54.39 I E 5..: Bill oows $ 38,. i ode to uy ew SUV. The 5 ye decliig-lce lo othe me fo lo tht is eig educed vi motiztio schedule hs tem of 7 yes. A Clculte the mothly pymet D, the sum of ll mothly pymets PV PS, d the sum of ll iteest pymets PV IPS. B Clculte D A d D I. C Fid the pymet ume J whee the mout eig pplied to picipl stts to eceed 9 of the pymet. A : D T { }.5 $38,. D.5 84 { } D $537.9mo : PV PV PV PS PS PS A TD 7 $539.9 $45,

159 3 : PV PV PV IPS IPS IPS TD A $45,5.43 $38,. $7,5.43 D A D A : DA $ $38,. B DA D $38.3 C A : D I : D : D Aj AJ J D D D A.467 A.9D D D A $ J J J I $ j.9d $ J l.467 l.737 J 6 l.737 l.467 J 58.7 Note: Notice the use of the tul logithm l whe solvig fo J. Tkig the logithm of oth sides is the stdd techique whe solvig lgeic equtios whee the vile ppes s epoet. I theoy, oe c use y se, ut l is stdd key ville o most scietific clcultos. 59

160 A iteestig questio ssocited with lo motiztio sks, wht pecet of the fist pymet is pplied towds picipl d wht pecet pys iteest chges? We ledy hve the lgeic mchiey i plce to swe this questio. To stt, D D Recll tht Aj A D A D A A T { }. D j Sustitutig the epessio fo D ito tht fo D A gives A A T { } D A A DA T A DA T Net, we fom the tio A D A D A T T { }. 6

161 Filly, we oti fte lgeic simplifictio DA D T E 3..3: Clculte D A fo 8.5 D ye d the followig vlues fot : 5,, d 3 yes. 5yes yes D D D D A A : 3yes D A T The epessio.9 9. : D D A 8.48 c e used to uild D lookup tle fo vious ul iteest tes d typicl lo motiztio tems whee the eties i the ody of the tle will e the coespodig picipl-to-ovell-pymet tios the vey fist motgge pymet. D A D fo ANNUAL INTEREST RATE TERM y y y y The ove tle helps swe questios such s, y wht pecetge would I hve to icese my mothly pymet i ode to educe my motiztio tem fom 3 yes to yes? If you motgge iteest te is 7, the swe fom tle lookup is ye oughly

162 3. You Home Motgge I his pop hit Phildelphi Feedom, Elto Joh sigs out the good old fmily home. The vst mjoity of ll Ameics puchse tht good old fmily home vi colltelized decliig-lce lo whee the colltel is the title deed to the house eig puchsed. This is the tditiol home motgge s we Ameics kow it. Two tems ssocited with the wod motgge e: motgge, the ledig istitutio gtig the motgge; d motggee, the idividul otiig the motgge. The esposiility of the motggee is to mke mothly pymets o time util tht time whe the lo is motized. I etu, the motggee is guteed plce to live i.e. the house cot e leglly esold o the motggee leglly evicted. Howeve, if the motggee fils to mke pymets, the the motgge c stt the legl pocess of evictio s mes of ecoveig the upid lce ssocited with the home motgge. Afte evictio occus, the ledig istitutio will sell the house, ecove the upid lce, 3 ecove epeses ssocited with the sle, d 4 etu y poceeds left to the motggee. The foemetioed sceio is ot hppy oe d should e voided t ll costs. Rememe, s log s thee is upid motgge lce, the ledig istitutio holds the title deed to the home tht you d you fmily occupy. Alwys mke sue tht the pymet you sig up fo is pymet tht you c cotiully meet moth fte moth d ye fte ye! The my emples i this ticle ddess vious spects of mkig motgge pymets d the totl lifetime costs ssocited with the motgge pocess. Let s egi with the most fequetly sked questio, how much is my pymet? E 3..: The Beett fmily is i the pocess of uyig ew home fo puchse pice of $ 3,.. They pl to put dow d fice the emide of the puchse pice vi covetiol fied-iteest-te home motgge with locl ledig istitutio. 6

163 The motiztio optios e s follows: T 5 ys@ 6.5, T ys@ 6.9 ye, d 3 ye T 3 ys@ 7.5 ye. Compute the mothly pymet fo ech of the thee optios. The iteest-te ge of. is fily typicl fo tem ye ge of 5 yes. The mout oowed will e $ 4,. fte the dow pymet is mde. Poceedig with the clcultios, we hve : T 5ys@ $6,. D.65 5 { } D $,9.3mo : T ys@ $6,. D.65 { } D $,98.3mo 3 : T 3ys@ $6,. D.75 3 { } D $773.66mo ye ye ye Of iteest would e the peset vlue PV PS TD of ll motgge pymets compisig the pymet stem fo ech of the thee optios. Oce PV PS is detemied, we c detemie PV IPS y the fomul PV IPS TD A. The esults fom E 3.. e show i the et tle 63

164 PRESENT VALUE FOR THREE PAYMENT STREAMS TERM PV PS A PV IPS 5 y $4,74. $6,. $4,74. y $53,59. $6,. $43,59. 3 y $638,57.6 $6,. $378,57. The fcts displyed i the ove tle e el eye-opee fo most of us whe fist eposed. The ottom lie is tht loge-tem motgges with lowe mothly pymets cost moe moey much moe moey i the log u. These cosidetios hve to e fctoed i whe uyig home. Sectio I: 6..8 lists some of the pos d cos ssocited with log-tem motgges. The et emple swes the questio, how much house c I ffod? E 3..: Bsed o icome, Bill Johso hs ee ppoved fo mothly motgge pymet ot to eceed $ 3. icludig el-estte tes d homeowes isuce. If, o the vege, el-estte tes e $ 4. pe ye d homeowes isuce is $ 6. fo homes i the sudivisio whee Bill wts to move, how much house c he ffod ssumig 3-ye motgge tes e 6.5? ye We e oly quotig the 3-ye te sice the ssocited motgge pymet will most likely e the lowest pymet ville. The motgge pymet tht icludes picipl, iteest, tes, d isuce is tditiolly kow s the PITI pymet, whees the pymet tht just icludes picipl d iteest is kow s the PI pymet. The fist step will e the sutctig out of the mothly potio of the $ 3. motgge pymet tht must e llocted to tes d isuce. $4. $6. : D $3 D $533.mo 64

165 I the secod step, we set $ 533. equl to the mothly pymet fomul d solve fo the ssocited motgge mout A..65 A :$ { }.65 3 { }{ $533.} A.65 A $4,8.74 I summy, Bill qulifies fo $ 4,. motgge. If oe ssumes tht Bill hs eough moey to mke dow pymet, the Bill would e qulified to uy house hvig puchse pice P P of $ 5,. s show i the lgeic clcultio elow. P P P P P P.P P $4,..8 $5,. $4,..8P P $4,. Notice tht the dow pymet eeded ude the ove sceio is hefty$,.. The et emple swes the questio, if I icese my pymet y so my dolls pe moth, how much sooe will I e le to py off my motgge? E 3..3: Nth d his wife Ncy puchsed house seve yes go, ficig $ 75,. fo 3 yes t 7 ye. The couple s mothly icome hs ecetly icesed y$ 5.. Nth d Ncy decide to use $ 5. of this icese fo dditiol mothly piciple pymet. A If the couple follows this pl, how my yes will they e le to sve fom the cuet 3 yes emiig o the motgge? B How much moey will they sve i iteest chges? 65

166 I Step, we clculte the eistig mothly pymet y the usul method. : T 3ys@ 7..7$75,. D.7 3 { } ye D $64.8mo I Step, we clculte the lce pyoff emiig o the motgge t the ed of seve yes. j D j : PO j A { } PO.7 $75, { } $ { 84 } PO $59, Keepig the sme pymet of D $64. 8mo llows the emiig piciple of $ 59, to e pid off i 3 yes ight o schedule. Icesig the pymet to D $44. 8mo will logiclly esult i compessio of the emiig tem. Ou ppoch fo the emide of the polem is to use the eistig mothly pymet fomul A 84 T { } D i evese i ode to solve fo T whe D, A, d is kow. Fist otice tht A D T { }.7$59,57.97 D.7 3 { } D $64.8mo. 66

167 The pevious esult cofims the powe of the eistig mothly pymet fomul i tht this fomul etis the lgeic likge mogst picipl, pymet, iteest te d tem t y stge i the motiztio pocess. It lso llows oe to solve fo y oe of the fou viles povided the othe thee viles e kow. With this i mid, we filly poceed to Step 3 whee D is icesed to D $44. 8mo. 3 : D T { }.7 T { }.7 T $6,97.36{ }.7 T { } T T l l.349 T T 5.36yes A.7$59,57.97 $44.8 T 5. 36yes $,65.56 The swe epesets 85 pymets whee the fil pymet is smll fctiol pymet tht would ceemoiously py off the motgge. Goig ck to the oigil questio, Nth d Ncy would compess the oigil motgge y 4 A :3.yes 5.36yes 7. 64yes y icesig the pymet to D $44. 8mo. To swe pt B, we clculte the oigil mout pogmmed to iteest ssumig the full thity-ye schedule d the eclculte it fo the mout ctully pid. The diffeece is the svigs. 67

168 5 : PV PV PV 6 B : PV oigil TD A $49,4.8 $75,. $44,4.8 eclculted 7$ $44.8 $75,. PV IPS IPS IPS IPS IPS eclculted $83,479.6 Svigs $6,66.9 Thus Nth d Ncy will e le to sve $ 6,66. 9 i iteest chges if they fithfully follow thei oigil pl. I the et emple, the motgge iitilly hs tem of 3 yes d the motggee wishes to motize it o cceleted ye schedule fte five yes hve elpsed i the oigil tem. E 3..4: Bi Smith puchsed house five yes go d ficed $ 5,. fo 3 yes t 7.. He would like ye to py off his house i 5 yes. A By how much should he icese his mothly pymet i ode to mke this hppe? B How much does he sve i the log u y followig the compessed epymet schedule? Step is the clcultio of the eistig mothly pymet. : T 3ys@ 7..7$5,. D.7 3 { } D $459.39mo ye I Step, we clculte the pyoff t the ed of five yes. 68

169 : PO PO PO A j { } j.7 $5, { } $ j.7 $,89.89 D 6 { 6 } Bi wts to ccelete the motgge epymet schedule so tht the emiig $, is pid off i 5 yes. This, i effect, cetes d ew 5-ye motgge hvig the sme ul iteest te. Step 3 is the clcultio fo Bi s ew pymet. 3 : T 5ys@ 7..7$,89.89 D.7 5 { } ye D $845.66mo Oce the old d evised pymets e kow, Pt A is esily sweed. 4 A : icese $ $ $ mo Pt B: Follow the ect pocess s peseted i Emple 5..3, Steps 5 d 6, to oti Bi s ovell pojected svigs of$ 5,748.. I ou et emple, motgge is iitilly tke out fo tem of yes. Thee yes ito the tem, the motgge is eficed i ode to oti lowe iteest te. E 3..5: I uyig ew home, the Pickles ficed $ 59,. fo yes t 6. ye. Thee yes lte, 5- ye tes dopped to ye. The Pickles decide to efice the emiig lce d the ssocited $ 5. eficig closig costs t the lowe te. How much do they sve ovell y completig this tsctio? 69

170 : T ys@ 6..6$59,. D.6 { } D $57.55mo : PO PO PO 3 A ye j { } j.6 $59, { } $ j.6 $45,74.48 D 36 { 36 } : T 5ys@ $47,4.48 D { } D $54.8mo ye Notice tht the mothly pymet ctully dops little it, d we hve compessed the ovell tem y two yes! Usig ou stdd methodology, the ovell svigs is 4 : 4$57.55 {36$ $54.8}. $8,74.9 Ou lst emple illusttes the devsttig cumultive effects of mkig ptil motgge pymets ove peiod of time. Hopefully, this is situtio tht most of us will stive to void. E 5..6: Tees ought ew home fo puchse pice of $ 45,.. She mde $ 9,. dow pymet d ficed the emide t 7 ye fo tem of 3 yes. Thee yes ito the lo, Tees ws cut to hlf-time wok fo peiod of 4 moths. 7

171 Tees ws le to egotite with he ledig istitutio ptil motgge pymet hlf the oml mout fo the sme peiod. At the ed of the 4 moths, Tees ws le to go ck to fulltime employmet d mke full house pymets. A Clculte he motgge lce t the ed of five yes. B Clculte the evised emiig tem if the oigil pymet is mitied. C Clculte the evised pymet eeded i ode to motize the lo vi the oigil schedule. Fist, we eed to clculte Tees s oigil pymet: : T 3ys@ 7..7$36,. D.7 3 { } D $395.9mo ye. At the ed of thee yes, the motgge lce is j D j : PO j A { } PO PO.7 $36, { $395.9} { 36 }.7 $348, We use the sme fomul the secod time i ode to clculte the effects of mkig mothly hlf pymet of $ fo peiod of two yes o ptilly-motized lo hvig sttig lce$ 348, j D j : PO j A { } PO PO.7 $348,7.3 { $97.54} { }.7 $369,

172 A Tees s evised motgge lce t the ed of five yes is $ 369,67.84, sum which is $ moe th she oigilly oowed. At the ed of five yes, the oigil pymet of $ comes ck ito ply, pymet tht must py off lce of $ 369,67.84 ove yet-to-e-clculted ume of yes. 4 A : D T { }.7$369,67.84 $ T { }.7 T $8,74.96{ }.7 T.7 { }.95.7 l l T T T yes $5, T.9975 B With the oigil pymet, Tees will ot py off he motgge util othe 33 yes hve pssed. Whe dded to the five yes tht hve ledy tspied, this motgge will equie 38 yes to motize ssumig o othe chges occu. To ig Tees ck o schedule, we will eed to clculte evised motgge pymet tht llows he to motize the lce of $ 369, i 5 yes. 5 : T 5ys@ 7..7$369,67.84 D.7 5 { } ye D $6.45mo C Tees s evised motgge pymet is $ 6.45mo, $ 37.36mo moe th he oigil pymet of $ Plyig ctch up is costly! 7

173 3.3 C Los d Leses Nowdys, most c los e set up o decliig-lce motiztio schedules. The mthemtics ssocited with c los set up o decliig-lce motiztio schedule is ideticl to the mthemtics ssocited with home motgges. Two mjo diffeeces e tht the tem is much shote fo c lo d tht the ul iteest te is ofte less. Let s stt off y computig c pymet. E 3.3.: Bo ought 4 SUV hvig sticke pice of $ 45,.. The slespeso kocked off, del tht Bo gldly geed too. Afte fctoig i 7 stte sles t o the geed-to sles pice, Ro put $. dow d ficed the lce fo 66 moths t 4 ye. The ledig istitutio hppes to e susidiy of the c mufctue. A Clculte Bo s c pymet. B Clculte the iteest pid to the ledig istitutio ssumig the lo goes full tem. A : Sles P ice.88 $45,. $39,6. Sles P ice T.7 $39,6. $4,37. AmoutFiced $4,37. : T 5.5ys@ 4..4$4,37. D.4 66 { } D $68.46mo ye 3 : PV 66$68.46 $4,37. B IPS PV $467.6 IPS 73

174 The fscitig thig out Emple 5.3. is tht totl iteest $ to e pid to the ledig istitutio pt of the c coglomete just out equls $ 54., the doll mout kocked off the oigil sles pice. Could this e clssic cse of py me ow o py me lte? A el dge i ficig lge mouts fo epesive vehicles is tht vehicles ulike houses depecite ove time. This mes tht thee my e peiod of time withi the tem of the lo whee the ctul lce emiig o the lo eceeds the cuet vlue of the vehicle itself. Such peiod of time is popely chcteized s ficil dge zoe sice isuce poceeds pid vi the totlig of fully-isued vehicle i the dge zoe will ot e eough to etie the ssocited lo. Thus, the oce poud owe is ot oly stuck with tshed vehicle, ut lso ptilly upid det d, most ssuedly, sigifictly highe isuce pemiums i the futue. Motoized vehicles, s much s Ameics love em, e defiitely mjo fmily moey di. So, y how much does vehicle typiclly depecite? The stdig ule of thum is etwee 5 d pe ye whee the sttig vlue is the mufctues suggested etil pice. The 5 figue is good ume fo highe-piced vehicles equipped with desile stdd optios such s i coditioig d utomtic tsmissio. The figue is usully eseved fo chepe stipped-dow models hvig few custome-eticig fetues. Eithe pecetge figue leds to simple mthemticl model desciig c depecitio. Let SRP e the suggested etil pice of pticul c model, P e the ssumed ul depecitio te s deciml fctio, d t e the ume of yes tht hve elpsed sice puchse. The the cuet vehicle t vlue V V t c e estimted y V t SRP P whee SRP is the mufctues suggested etil pice; P is the ul depecitio te; t e the ume of yes sice puchse. 74

175 Note: Some estimtos sy tht oe must immeditely educe vehicle s vlue fom esle vlue to wholesle vlue s soo s it leves the showoom. Tht mout is oughly equivlet to oml ye s depecitio, which iceses the epoet up y t oe i the pevious model V t SRP P. E 3.3.: Poject the vlue of Bo s SUV ove the life of the coespodig lo with d without immedite Showoom Depecitio. Use ul depecitio te of P. 5 d clculte the two vlues t si-moth itevls. Lookig ck t the pevious emple, we see tht SRP $45,.. The esults otied vi the two vehicledepecitio models e show i the tle elow. Time i moths DEPRECIATION OF BOB S SUV With Showoom Depecitio Without Showoom Depecitio $38,5. $45,. 6 $35,64. $4,487. $3,5. $38,5. 8 $9,975. $35,64. 4 $7,635. $3,5. 3 $5,478. $9, $3,49. $7, $,656. $5, $9,966. $3, $8,48. $, $6,97. $9, $5,647. $8,48. Oe c gue out with o without showoom depecited, ut eve with o depecitio, Bo s SUV dops out $ 35. of its sticke pice i the fist si moths. The impott thig to ote is tht the tle vlues e the isuce vlue of the vehicle i.e. the csh tht isuce compy will py you if the vehicle is totlly destoyed. Yes, you my e le to sell it fo moe; ut wht if it is ivolved i ccidet? The tle vlue will e you legl compestio. 75

176 Let s see how Bo s SUV lo pogesses towds pyout duig the sme 66-moth tem. We will compute the emiig lce t si-moth itevls usig the ow-fmili fomul The esults e: j { } j D PO j A. AMORTIZATION OF BOB S SUV LOAN Time i moths Remiig Lo Blce $4,37. 6 $37,57.6 $33, $3, $6,76. 3 $3, $9, $5, $, $8,5.6 6 $4, $.6 Note the few cets emiig o the lo lce. Icesig the mothly lo pymet to eve $ 683. will esily elimite tht polem cused y oudig eos ppoch most ledig istitutios would tke. Now fo the momet of tuth! We will mege the lst two tles ito ew tle i ode to compe depecited vlue to cuet lo lce lie-y-lie. 76

177 Time i moths BOB S SUV LOAN, A LOAN ON THE EDGE! With Showoom Depecitio With No Showoom Depecitio Remiig Lo Blce $38,5. $45,. $4,37. 6 $35,64. $4,487. $37,57.6 $3,5. $38,5. $33, $9,975. $35,64. $3, $7,635. $3,5. $6,76. 3 $5,478. $9,975. $3, $3,49. $7,635. $9, $,656. $5,478. $5, $9,966. $3,49. $, $8,48. $,656. $8,5.6 6 $6,97. $9,966. $4, $5,647. $8,48. $.6 The ove tle shows lo o the edge! If we fcto i showoom depecitio, the isuce vlue of the vehicle is ctully less th the lce emiig o the lo fo out the fist two yes. We could tem tht peiod of time ficil dge zoe sice the isuce poceeds fom totled vehicle will ot e eough to py off the lo i full. If we do t fcto i showoom depecitio, we e i esoly good shpe thoughout the sme two yes ig if. So, we might coclude tht Bo is ot i too get of dge. But, how out M. Hvey, whose stoy is i ou et emple. E 3.3.3: M. Roet Hvey ought ew Cmy fo his so Joh, who pled to use it while ttedig college. The oigil Cmy sticke pice of $ 4,995. ws discouted y$ 5. due to Toyot dvetised sle. Stte d couty sles tes the dded 6 to the emiig puchse pice. M. Hvey mde $. dow pymet d ficed the lce fo five yes t3.5, figuig the c would e pid off whe Joh gduted. ye Als, fte hd diffeet pl ecuse poo Joh totled it sevetee moths lte. Poject the upid lo lce, if y, fte isuce poceeds e eceived. 77

178 : Sles P ice $4,995. $5. $3,495. Sles P ice T.6 $3,495. $4,94.7 AmoutFiced $4,94.7 $,. $3,94.7 : T 5ys@ $3,94. D.35 6 { } D $434.85mo ye At the sevetee-moth poit, we eed to clculte oth the emiig wholesle vlue of the Cmy which hopefully equls the isuce poceeds d the emiig lce o the lo. Also, s ule, the Toyot Cmy holds its esle vlue the well. Thus, we will e optimistic d use P. 3 i cojuctio with showoom depecitio. Notice the esclig of the time t to moths. 3 : V t SRP P V t $4, V t $7, : PO PO $ PO $7, j A j { } j $3,94. D { } : SettlemetBlce $7,85.8 $7,549.8 SettlemetBlce $3.36 t 9 78

179 M. Hvey escped y the ski of his teeth. Afte the lo lce is pid off, he will hve pocketed$ But wit, M. Hvey will hve to come up with dditiol dow pymet ecuse Joh ow eeds othe c. Life o the edge! The lst stoy might hve ee sigifictly diffeet if othe model of utomoile ws ivolved. Let s ssume tht the puchse pice, discout, tes, d lo coditios emi ideticl ut the mke d model of c is oe fo which P.. The, sttig gi t Step 3, we hve 3 : V t SRP P V t $4, V t $4, : PO 5 7 $7,549.8 t 9 : SettlemetBlce $4,576.5 $7,549.8 SettlemetBlce I this sceio, M. Hvey still owes $ to the ledig istitutio oce isuce poceeds e eceived. Plus, he ll eed some dditiol csh fo ew dow pymet o eplcemet vehicle. Hece, y sigig o to this del, M. Hvey olled o the edge d evetully fell off. Ou et emple is tke fom dvetisemet i locl ewsppe. E 3.3.4: A Fod deleship is dvetisig d ew 4 Feest fo sles pice of $ 7,483., which is $ 5. less th the mufctues suggested etil pice. Fod will fice the whole mout with othig dow fo qulified uyes fo 84 moths t 5.89 ye. The dvetised pymet is $ 69.mo. Alyze this del fo coectess, tue cost d edgiess.. 79

180 We fist eed to dd i the 7 Stte-of-Ohio sles t to get the tue mout ficed; the, we compute the mothly pymet. : Sles P ice $7,483. Sles P ice T.7 $7,483. $8,76.8 AmoutFiced $8,76.8 : T 7ys@ $8,76.8 D { } D $7.9mo ye Notice tht we e oly out $ 3. wy fom the dvetised pymet; hece we will ccept the deleship s clcultios s vlid. Note: the smll diffeece is poly due o how we itepeted the stted te of 5.89 s eithe effective ul te ye eff o ctul ul te. Net, let s compute the sum of ll iteest pymets duig the life of the lo. 3 : PV PV PV IPS IPS IPS TD A 84 $7.9 $8,76.8 $ A impott thig to ote hee is tht the deleship is giig ck 8 of the dvetised ete $ 5. i iteest chges. The hook is the lue of o moey dow. Lstly, let s emie lo edgiess i tems of emiig lo lce vesus the depecited vlue of the Feest. Cosideig the size of the iitil ete, ssume tht the iitil showoom discout hs ledy occued. 8

181 Hece, the ppopite depecitio model is t V t SRP P ; d, sice the Feest hs desile fetues, we will use P. 5. Tle 5.7 shows the fightful esults Feest o the edge fo ely fou yes! Time i moths A FREESTAR ON THE EDGE With o Showoom Depecitio Remiig Lo Blce $7,483. $8, $6,8.5 $7,6.6 $4,86.55 $6, $3,7.75 $5, $,63.47 $4,.99 3 $, $, $, $, $9, $, $9,6.3 $8, $8,43.97 $7, Ou lst emple i this sectio emies vehicle lese. A lese is lo tht fices the coespodig mout of vehicle depecitio tht tspies duig the tem of the lo. At the ed of the peiod, the vehicle is etued to the deleship. All leses hve stipultios whee the mout of miles ggegted o the vehicle must emi elow usully, miles pe ye. E 3.3.5: A Gd Cheokee is dvetised fo ed tg sles pice of $,888. fte etes. The coespodig ed-tg lese pymet is $ 48.mo plus t fo tem of 39 moths with $ 999. due t sigig. Fom the ifomtio just give, lyze this tsctio. The sles pice of $,888. epesets out off d my ctully e little it elow wholesle. But, wht does it mtte, fo the vehicle is goig to evetully e etued to the deleship d esold s pemium used c! 8

182 Pedictig the oigil mufctues suggested etil pice SRP, we hve :.8 SRP $,888. SRP $7,36. Net, we pedict the depecitio duig the 39 moth tem of the lese usig the showoom depecitio model with P. 5. t : V t SRP P 5 V 39 $7, V 39 $3,73.44 Oce the depecitio is clculted, we c detemie the ctul mout ficed d the iteest chged. 3 : AF $,888. $3,73.44 $999. AF $ $999. $ : PV 5 : PV PS IPS 39 $48. $967. $967. $ $ The diffeece PVIPS is due to the pplied iteest te ove the tem of 39 moths, which we will ow detemie y: 6 : FV $ l l PV 39 l T PV $969. T 39 FV Notice the sky-high iteest te of 9., te tht is ppochig low-ed cedit-cd tes! I closig Aticle 3.3, we will leve it to the ede to veify the followig sttemet: To void livig o the edge whe sigig up fo utomoile lo, mke dow pymet equivlet to the fist ye s depecitio, icludig showoom depecitio.. 8

183 3.4 The Auity s Motgge i Revese A uity c e thought of s motgge i evese whee the uitt the oe eceivig the pymet ecomes the lede d the istitutio fom which the uity is puchsed ecomes the oowe. Thus, mothly uity pymets e computed vi the sme methods used fo computig mothly motgge pymets. With the lst sttemet i mid, we poceed with just oe compehesive emple tht ddesses oth uity cetio d uity usge. E 3.4.: Mike, ge 5, eceives $,. s iheitce. Usig his iheitce moey s iitil deposit, Mike wisely decides to ope compy-sposoed 4K ccout. Fo 4 yes, he mkes ul pyoll deposit of $. which the compy mtches. A Poject the vlue of Mike s 4K ccout t ge 67 ssumig vege effective ul te of etu of eff 9 ye. B If the totl vlue i Mike s 4K ccout is used to uy thity-ye-fied-pymet uity pyig 5 ye t ge 67, clculte Mike s mothly etiemet pymet. C If Mike dies t ge 87, how much is left i his 4K ccout? A Auity Cetio Phse Step is the costuctio of moety-gowth digm. t eff 9 :$,. 4 ye $ 4. t FV? Step is pojectig the Futue Vlue of Mike s 4K t D t : FV4K LS D eff eff FV FV 4K 4K 4 { } 43 { } 4 $4. $ $3,95.9 $,763,38.65 $,987,87.84 eff 83

184 B Auity Pymet Phse Usig the fomul A T { } D eeded to motize motgge, we oti D.5$,987, { } $8,8.36 D.7767 D $,668.8mo fo mothly pymets C Blce Left i Auity t the Ed of Yes j D j PO j A { } PO PO.5 $,987,87.84 { $,668.8} { }.5 $,5,85.89 Whe Mike dies t ge 87, he leves $,5, i oliquidted fuds. Hopefully his uity is such tht y uused mout evets to Mike s esttes d heis s specified i will. 4 84

185 4. The Clculus of Fice 4. Jco Beoulli s Diffeetil Equtio A questio commoly sked y those studets stugglig with equied mthemtics couse is, Wht is this stuff good fo? Though sked i evey mthemtics couse tht I hve tught, I thik usiess clculus is the oe couse whee this questio equies the stogest espose. Fo i my othe clsses pe-lge, lge, etc. I c gue tht oe is leig uivesl lguge of qutifictio. Susequetly, to essetilly sk of wht good is this lgeic lguge? is to miss the whole poit of hvig ville ew, poweful, d ect mes of commuictio. To ot hve this commuictio mes t my disposl could e likeed to ot eig le to spek Eglish i pimily Eglish-spekig couty. To sy tht this would e hdicp defiitely is udesttemet! Yet this is pecisely wht hppes whe oe does t spek mthemtics i techologicl wold ulig ove with mthemticl lguge: e.g. umes, dt, chts, d fomuls. I hve foud though epeiece tht the pevious gumet mkes good cse fo pelge d lge; howeve, mkig simil cse fo usiess clculus my equie moe specifics i dy whe Micosoft EXCEL ules. I this ticle, we will eploe oe vey essetil specific i the mode wold of fice, mely the gowth d decy of moey y the use of diffeetil equtios, oe of the lst topics ecouteed i stdd usiess clculus couse. Jco Beoulli ws estled i etwee the lifetimes of Leiiz d Newto, the two co-foudes of clculus. Jco ws out yes youge th eithe of these me d cotiued the tditio of stdig o the shouldes of gits. Oe of Jco s getest cotiutios to mthemtics d physics ws mde i the ye 696 whe he foud solutio to the diffeetil equtio elow, which es his me. dy f y g y d Of pticul iteest i this ticle is the cse fo : 85

186 86 g y f d dy. The solutio is otied vi Beoulli s 3-ye-old methodology s follows. Step: Let F e such tht f F Step : Fomulte the itegtig fcto F e Step 3: Multiply oth sides of g y f d dy y F e to oti [ ] ] [ g e y f e dy dy e g y f e d dy e F F F F F Whee the left-hd side of the lst equlity is the deivtive of poduct [ ] y e d d g e y f e dy dy e F F F F ] [. Step 4: To complete the solutio, pefom the idefiite itegtio. [ ] [ ] F F F F F F F Ce d g e e y y C d g e y e g e y e d d

187 4. Diffeetils d Iteest Rte Eveyoe will gee tht fied mout of moey p will chge with time. Eve though p $,. is stuffed ude mttess fo twety yes i the hopes of pesevig its vlue, the pssge of twety yes will chge p ito somethig less due to the eve-peset ctio of ifltio deoted y i i this ticle, which c e thought of s egtive iteest te. So popely, p pt whee t is the idepedet vile d p is the depedet vile. Let dt e diffeetil icemet of time. Sice p pt, dt will iduce coespodig diffeetil chge dp i p vi fist-ode lie epessio likig dp to dt : dp Kdt dp K t dt. The ect fom of the popotiolity epessio K t will deped o whethe piciple is gowig, decyig, o whethe thee is ume of complemety d/o competig moety-chge mechisms t wok. Ay oe of these mechisms my e time depedet i d of itself ecessittig the witig of K s K Kt. The simplest cse is the moety gowth mechism whee K p, the poduct of costt iteest te d the iitil piciple p. This implies costt te of doll icese with time fo give p, which is the tditiol simpleiteest gowth mechism. Thus dp p : p. dt p The pecedig is fist-ode lie diffeetil equtio witte i septed fom with stted iitil coditio. It c e esily solved i thee steps: 87

188 : p t p : p p 3 : p t p t C C p t p p t Oe might ecogize the lst epessio s the fuctiol fom of the simple iteest fomul. The sme diffeetil equtio c e witte s dp dt p : p fte divisio y dt. p This fom highlights the diffeetil-sed defiitio of the fist deivtive. I wods it sttes tht the tio of iduced diffeetil chge of piciple with espect to coespodig, itisic diffeetil chge i time is costt, eig equl to the pplied costt iteest te times the iitil picipl, lso costt. Simple emitio of oth sides of the ove diffeetil equtio evels commo d cosistet uits fo oth sides with dp dt dolls dolls & p. ye ye dp The epessio p t is kow s the Leiiz fom of the dt fist deivtive, equl to the istteous chge of piciple with espect to time which oe could immeditely like to istteous velocity of moey gowth. 4.3 Beoulli d Moey dp K t dt Retuig to, we hve fo the geel cse tht K t t p t d t whee t is time-vyig vile iteest te, pt is the picipl cuetly peset, d dt is idepedet vile deposit te.. 88

189 Sustitutig ito dp K t dt gives o [ t p t d t ] dt : p p dp dp dt t p t d t : p p whee p p is the mout of picipl peset t the oset of the pocess. Tsltig the diffeetil equtio ito wods, the istteous te of chge of picipl with espect to time equls the sum of two idepedetly ctig qutities: the poduct of the vile iteest te with the picipl cocuetly peset d vile diect-dditio te. The pecedig diffeetil equtio is pplicle i the usiess wold if the picipl p is cotiuously gowig o decliig with time. Whe the iteest te is fied t d the idepedet diect-dditio te is zeo d t, the diffeetil equtio educes to dp dt : p. p p Solvig usig septio of viles gives dp : dt p : l p t C p t e 3 : p p p t p e t The fil epessio p t pe is the fmili Cotiuous- Iteest Fomul fo piciple gowth give sttig picipl p d costt iteest te. t C e t. 89

190 Retuig to the geel diffeetil equtio dp dt t p t d t : p p, we see tht it is Beoulli i fom with the solutio give gi y tocious epessio F t p t e t dt F t F t F t [ e d t dt] Ce Upo compiso with the geel solutio developed i detil elie. The iitil coditio p p will e pplied o csey-cse sis s we eploe the vious d poweful uses of the ove solutio i the wold of fice. Depedig o the compleity of t d d t, the coupled solutio F t p t e t dt F t F t F t [ e d t dt] Ce : p p my o my ot e epessile i tems of simple lgeic epessio.. Thus, sice iteest tes e upedictle d out of y oe idividul s cotol I hve see doule-digit swigs i oth svigs-ccout tes d motgge tes i my lifetime, we will ssume fo the pupose of pedictive lysis tht the iteest te is costt thoughout the time itevl of iteest t. This immeditely leds to [ e ] t t d t dt Ce : p t p t e p, cosidele simplifictio. The lst esult is ou sttig poit fo cocete pplictios i ivestmet plig, motgge lysis, d uity plig. 9

191 4.4 Applictios 4.4. Gowig Nest Egg Cse : If d t d, costt ul deposit te, the the lst epessio fo pt futhe simplifies to [ e ] t t dt Ce : p p t d p. e t This c e esily solved to give t d t p t pe [ e ] fte pplyig the oudy coditio p p. Notice tht the ove epessio cosists of two distict t tems. The tem pe coespods to the picipl ccued i cotiuous iteest-eig ccout ove time peiod t t costt iteest te give iitil lump-sum ivestmet p. d t Likewise, the tem [ e ] esults fom diect picipl dditio vi ul meteed cotiutios ito the sme iteesteig ccout. If eithe of the costts p o d is zeo, the the coespodig tem dops wy fom the ovell epessio. The followig two-stge ivestmet polem illusttes the use of t d t p t pe [ e ]. E 4.4.: You iheit $,. t ge 5 d immeditely ivest $,. i copote-od fud pyig 6 ye. Five yes lte, you oll this ccout ove ito solid stock fud whose fifty-ye vege is 8 d stt cotiutig $ 3. ully. A Assumig cotiuous d stedy iteest, how much is this ivestmet woth t ge 68? B Wht pecet of the fil totl ws geeted y the iitil$,.? ye 9

192 A I the fist five yes, the oly gowth mechism i ply is tht iduced y the iitil ivestmet of $,.. Thus, the mout t the ed of the fist five yes is give y.65 p 5 $,.e $3, The output fom Stge is ow iput to Stge whee oth gowth mechisms ct fo dditiol 38 yes p38 3,498.58e e.8 p38 $48,797. $375,869. p38 $58, B The of the fil totl ccued y the iitil $,. is $48, $58, Note: The iitil ivestmet of $,. is geetig 8. of the fil vlue eve though it epesets oly 8 of the ovell ivestmet of $ 4,.. The elie lge sum of moey is iheited o eceived y idividul, the wise it eeds to e ivested; d the moe it couts lte i life. Holdig the ul cotiutio te to $ 3. ove peiod of 38 yes is ot elistic thig to do. As icome gows, the coespodig ul etiemet cotiutio should lso gow. Oe mthemticl model fo this is dp dt αt p de p : p whee the costt ul cotiutio te d i the pevious α t model d hs ee eplced with the epessio de, llowig the ul cotiutio te to e cotiuously compouded ove time peiod t t vege ul gowth te α. 9

193 The ove equtio is yet othe emple of solvle Beoulli-i-fom diffeetil equtio pe the sequece p t e t p t d e p t p e o t αt [ e de dt] t α t [ e dt] t Ce Ce t t d [ t αt e e ] α : p p : p p E 4.4.: Repet E 4.4. usig the ul cotiutio model.3t d t 3e. A Stge emis the sme with p 5 $3, The Stge clcultio ow ecomes p38 3,498.58e e e.8.3 p38 $48,797. $,66,78.49 P38 $,5, The fil ul cotiutio is $3.e $ with the totl cotiutio thoughout the 38 yes is give y the defiite itegl 38 $3.e.3t $,.e dt.3t 38 $, B The of the fil totl ccued y the iitil $,. is $48,79... $,5,5.7 Most of us do t eceive lge mout of moey ely i ou lives. Tht is the eso we e tio pimily mde up of middle-clss idividuls. So with this i mid, we will fogo the ely iheitce i ou et emple... 93

194 E 4.4.3: Assume we stt ou ivestmet pogm t ge 5 with ul cotiutio of $ 3. gow t te of α 5 pe ye. Also ssume ggessive ul iteest te of epets tell us tht this is still dole i the log tem though smt ivestig. A How much is ou est egg woth t ge 68? B How does ssumed vege ul ifltio te of 3 thoughout the sme time peiod lte the fil esult? A Diect sustitutio gives 3 p43 e..5 p43 $3,96, e.543 B Ifltio is othig moe th egtive gowth te o iteest te tht deits the give te. Fo 3 vege ul ifltio te, the tue iteest T d icome gowth tes α T e give y the two epessios T i α α i 5 3. T Sdly, ou tue vlue fte 43 yes i tems of tody s uyig powe is 3 p43 e.7. p43 $,75, Pyig fo the Nest.743 e.43 Both motgge los d uities e, i ctulity, ivestmet pls i evese whee oe stts with give mout of piciple p p d chips wy t this iitil mout util tht poit i timet whe p T. The goveig equtio fo the cse whee the iteest te is fied thoughout the motiztio peiod T is 94

195 t [ e ] t d p t pe whee d ow ecomes the equied ul pymet. Applyig the coditio p T leds to t pe e d. t The fied mothly pymet m is give y m d p e t t { e } The cotiuous-iteest-picipl-eductio model does ecellet jo of clcultig ely-coect pymets whe the ume of compoudig o picipl eclcultio peiods eceeds fou pe ye. Below e thee othe motgge-pymet fomuls sed o the cotiuous-iteest model. Fist Moth s Iteest: Totl Iteest I Pymet : Totl Amout Pid p T Te I p T e T Tpe A p I : A T e E 4.4.4: $ 5,. is oowed fo 3 yes t Clculte the mothly pymet, totl epymet, d totl iteest epymet ssumig o ely pyout. m.575$5,. e.5753 e $5,. e A.5753 e.5753 $457.6 $54,

196 I A p $54,745.5 $5,. $74,745.5 My people justify iitilly-high motgge pymet due to the fct tht the motgge is eig pid off i chepe dolls. This sttemet efes to the effects of ifltio o futue motgge pymets. Futue motgge pymets e simply ot woth s much i tody s tems s cuet motgge pymets. I fct, if we poject t yes ito the lo d the cotiuous ul ifltio te hs ee i thoughout tht time peiod, the the peset vlue of ou futue pymet P e T i t m PV e T. e mpv To illustte usig E 4.4.4, the peset vlue of pymet mde yes fom ow, ssumig i 3 ye is m PV.3 $457.6e $ Thus, ude stle ecoomic coditios, ou ility to comfotly ffod the motgge should icese ove time. This is cse whee ifltio ctully woks i ou fvo. Cotiuig with this discussio, if we e pyig off ou motgge with chepe dolls, the wht is the peset vlue of the totl mout pid A PV? A simple defiite itegl itepeted s cotiuous summig povides the swe T T T i T P e i t P e e A PV e dt T T e i e Retuig gi to E 4.4.4, the peset vlue of the totl 3-ye epymet stem is A $345, PV is 96

197 A PV E 4.4.5: Compe m, A, d fo motgge whee p $3,. if the fied iteest tes e: 3 6, yes 5.75, d 5 yes 5. ifltio te of 3 yes. Assume stedy ul i d o ely pyout. I this emple, we dispese with the clcultios d peset the esults i the tle elow. FIXED RATE MORTGAGE COMPARISON FOR A PRINCIPAL OF P $3,. Tems M A A PV T 3 6. $797.5 $646,938. $46,569.6 T 5.75 $3.57 $54,856.8 $ T 5 5. $369.9 $46,436. $343,396.6 The tle defiitely shows the mied dvtges/disdvtges of choosig shot-tem o log-tem motgge. Fo fied picipl, log-tem motgges hve lowe mothly pymets. They lso hve much highe ovell epymet, lthough the totl epymet is dmticlly educed y the ifltio fcto. The motgge decisio is vey much idividul oe d should e doe cosideig ll the fcts withi the scope of the ode ecoomic pictue. E 4.4.6: Ou lst emple is uity polem. Auities e simply motgges i evese whee mothly pyouts e mde, isted of mothly pymets, util the picipl is educed to zeo. You etie t ge 68 d ivest moey eed vi E i uity pyig 4.5 ye to e motized y ge 9. Wht is the mothly pyout to you i tody s tems? The phse, i tody s tems, mes we let p p 35. Thus, PV $,75, $,75, e m $6, e 97

198 The mothly icome povided y the uity looks vey esole efeecig to the ye 5. But, ufotutely, it is fied-icome uity tht will cotiue s fied fo 4 yes. Ad, wht hppes duig tht time? Ifltio! To clculte the peset vlue of tht mothly pymet, sy t ge 84, ou ow well-kow ifltio fcto i 3 is used to oti ye.36 m $6,6.79e $ I coclusio, the powe povided y the techiques i this shot sectio o fice is othig shot of miculous. We hve used Beoulli-i-fom diffeetil equtios to model d solve polems i ifltio, ivestmet plig, d istllmet pymet detemitio whethe los o uities. We hve lso evised the itepettio of the defiite itegl s cotiuous sum i ode to oti the peset vlue of totl epymet stem my yes ito the futue. These ecoomic d pesol issues e vey much tody s issues, d clculus still vey much emis wothwhile tool-of-choice eve fo mude ethoud polems some 3 yes fte its iceptio. 98

199 Appedices 99

200 A. Geek Alphet GREEK LETTER Uppe Cse Lowe Cse Α α Alph Β β Bet Γ γ Gmm δ Delt Ε ε Epsilo Ζ ζ Zet Η η Et Θ θ Thet Ι ι Iot Κ κ Kpp Λ λ Lmd Μ µ Mu Ν ν Nu Ξ ξ Xi Ο ο Omico Π π Pi Ρ ρ Rho Σ σ Sigm Τ τ Tu Υ υ Upsilo Φ φ Phi Χ χ Chi Ψ ψ Psi Ω ω Omeg ENGLISH NAME

201 B. Mthemticl Symols SYMBOL MEANING Plus o Add - Mius o Sutct o Tke Awy Plus o Mius do oth fo two ± esults Divide / Divide Multiply o Times ^ Powe isig Scl poduct of vectos { }o[ ]o Petheses Is equl to Is defied s Does ot equl Is ppoimtely equl to Is simil too > Is gete th Is gete th o equl to < Is less th Is less th o equl to, t, etc. Viles o poumes f o y Fuctio of idepedet vile Appoches limit d, dt, dy, etc. diffeetils f o y Fist deivtive of fuctio f o y Secod deivtive of fuctio

202 SYMBOL MEANING, Step, Step, etc. A B A implies B A B B implies A A B A implies B implies A! Fctoil i i Sig fo Summtio sig summig tems Sig fo idefiite itegtio o tidiffeetitio Sig fo defiite itegtio Poduct sig multiplyig tems Sig fo sque oot th oot Ifiity symol o the pocess of cotiuig idefiitely i like fshio Pllel Pepedicul Agle Right gle U I Tigle Set uio Set itesectio

203 SYMBOL MEANING A Memeship i set A A No-memeship i set A A B Set A is cotied i set B A B Set A is ot cotied i set B φ The empty set π e ϕ QED: thus it is show Fo evey Thee eists The ume Pi such s i 3. The ume e such s i.7 The Golde Rtio such s i.6 3

204 C. My Most Used Fomuls Fomul Pge Ref

205 5